ASTRONOMY 

A  POPULAR  HANDBOOK 


HAROJ  D  JACOBY 


•&RARY 

i        K1VERSJTY  OF 
V     CALIFORNIA 


ASTRONOMY 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •    CHICAGO  •   DALLAS 
ATLANTA  •    SAN   FRANCPSCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


Photo  by  Barnard,  Oct.  19,  1911.     Exposure,  46  min.  (see  p.  14). 

PLATE  1.     Comet  c  1911,  discovered  by  Brooks. 


ASTRONOMY 


A  POPULAR   HANDBOOK 


BY 

HAROLD   JACOB Y 

RUTHERFURD    PROFESSOR    OF    ASTRONOMY 
IN    COLUMBIA    UNIVERSITY 


WITH   THIRTY-TWO  PLATES  AND  MANY  FIGURES 
IN   THE   TEXT 


got* 

THE    MACMILLAN    COMPANY 
1915 

All  rights  reserved 


COPYRIGHT,  1913, 
BY  THE  MACMILLAN  COMPANY. 

Set  up  and  electrotyped.     Published  September,  1913      Reprinted 
August,  1915. 


Norfoooto 

J.  8.  Cashing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


915 


ASTRONOMY 
LIBRARY 


«  To  know 

That  which  before  us  lies  in  daily  life 
Is  the  prime  wisdom." 

—  Paradise  Lost,  VIII,  192. 


502 


PREFACE 

THE  present  volume  has  been  prepared  with  a  double 
purpose,  and  upon  a  plan  somewhat  unusual.  First,  an 
effort  has  been  made  to  meet  the  wishes  of  the  ordinary 
reader  who  may  desire  to  inform  himself  as  to  the  present 
state  of  astronomic  science,  or  to  secure  a  simple  explanation 
of  the  many  phenomena  constantly  exhibiting  themselves 
in  the  universe  about  him ;  and  the  further  purpose  has  been 
to  produce  a  satisfactory  textbook  for  use  in  high  schools 
and  colleges. 

Thus,  for  the  general  reader,  it  has  been  thought 
necessary  to  eliminate  all  formal  mathematics ;  for  the 
student,  on  the  other  hand,  the  occasional  use  of  elementary 
algebra  and  geometry  are  essential.  To  satisfy  these  two 
apparently  contradictory  conditions,  the  book  has  been 
written  in  two  parts ;  the  first  free  from  mathematics,  the 
second  a  series  of  extended  elementary  mathematical  notes 
and  explanations  to  which  appropriate  references  are  made 
in  the  first  part  of  the  book.  Thus  the  general  reader  may 
confine  his  attention  to  the  non-mathematical  part ;  the 
student  should  master  the  whole  volume. 

Attention  is  directed  especially  to  Chapter  I,  in  which  is 
presented  a  brief  summary  of  the  entire  science.  It  is 
hoped  that  this  will  serve  to  strengthen  in  most  readers  a 
desire  for  further  and  more  detailed  information.  To  the 
student  this  chapter  should  furnish  as  much  knowledge 
as  he  must  have  in  his  possession  before  beginning  a  direct 

vii 


PREFACE 

study  of  the  sky  with  a  telescope.  In  the  author's  extended 
experience  as  a  college  teacher  of  elementary  astronomy,  he 
has  found  it  most  desirable  to  give  life  to  the  subject  by 
requiring  frequent  evening  visits  of  students  at  the  obser- 
vatory. These  should  begin  almost  immediately  upon 
commencing  the  study  of  the  science ;  and  the  first  chapter 
is  therefore  intended  to  give  the  -students  something  to  work 
upon,  even  in  their  earliest  observatory  visits.  At  Columbia 
and  Barnard  colleges,  these  visits  are  required  on  frequent 
dates,  regularly  assigned  throughout  the  year,  without 
regard  to  the  state  of  the  weather.  When  clear,  the  tele- 
scope is  used ;  when  this  is  impossible,  oral  and  informal 
discussion  takes  place  upon  the  work  done  in  the  classroom. 
Attention  is  also  given  to  the  daylight  study  of  solar 
shadows,  all  students  being  required  to  construct  a  prac- 
tical sundial,  as  explained  in  Chapter  V. 

The  author  has,  of  course,  drawn  freely  upon  many  other 
books,  especially  in  the  preparation  of  numerous  diagrams, 
and  in  arranging  the  various  parts  of  the  subject  in  order. 
But  most  of  the  diagrams  are  new,  and  all  have  been  sim- 
plified as  much  as  possible.  In  a  few  cases,  illustrations 
were  copied  from  very  old  astronomic  textbooks:  references 
are  then  always  given,  in  the  hope  that  some  readers,  at 
least,  will  be  led  to  examine  these  fine  venerable  classics  of 
the  science. 

Almost  all  the  inserted  plates  are  photographic  repro- 
ductions of  actual  photographs.  For  these  the  author  is 
under  deep  obligations  to  Professor  E.  E.  Barnard,  of  the 
Yerkes  Observatory,  and  to  the  astronomers  of  the  Lick 
Observatory. 


COLUMBIA  UNIVERSITY, 
May,  1913. 


Vlll 


TABLE   OF   CONTENTS 

)H AFTER  PAGE 

I.     THE  UNIVERSE    .        ....        .        .         .        f    '     .         1 

Introductory.  General  view  of  the  science.  Its  practical 
use :  navigation,  coast  and  boundary  surveying,  timekeeping 
for  mankind.  Value  as  a  culture  study. 

II.     THE  HEAVENS    .        ,        .        .        .        ,        .        .        .        .      22 

What  we  can  see  by  examining  the  sky  without  a  tele- 
scope. The  celestial  sphere  with  its  points,  lines,  and  cir- 
cles. Diurnal  phenomena;  day  and  night;  rising  and  setting 
of  the  stars.  Aspect  of  the  heavens  from  New  York,  from 
the  equator,  and  from  the  polar  regions. 

III.  HOW    TO    KNOW    THE    STARS  .  .  .  ....         45 

The  planets  and  the  principal  fixed  stars  and  constella- 
tions. Maps,  globes,  and  planispheres. 

IV.  TIME     .        .        .        .        .        .        .        .  "      .        .        ..       .       65 

Star-time  or  sidereal  time.     Solar  time  and  standard  time. 
Differences  of  time  between  different  places  on  the  earth. 
The  international  date  line. 
V.     THE  SUNDIAL      .        .        .        .        .        .        ...        .      78 

How  to  make  one,  and  how  to  use  it. 

VI.     MOTHER  EARTH          ..        .        .        ...       ...        .        .       86 

Notions  of  the  ancients.  Proof  of  curvature  and  rotation. 
The  Foucault  experiment.  Measurement  of  the  earth's  size 
and  shape ;  by  the  ancients,  and  by  the  moderns.  Geodesy. 
The  earth's  mass:  weighing  the  earth.  Experiments  of 
Maskelyne  and  Cavendish.  The  terrestrial  interior.  Varia- 
tion of  latitudes.  The  atmosphere  :  twilight ;  refraction. 

VII.     THE  EARTH  IN  RELATION  TO  THE  SUN  .        .        .        .        .     116 

Orbit  around  the  sun  :  how  it  might  be  determined.  The 
seasons.  Astronomic  explanation  of  the  geologic  ice  age. 
The  length  of  the  year  determined  by  the  ancients.  Trop- 
ical and  sidereal  years.  Precession  of  the  equinoxes.  Nu- 
tation. Age  of  the  great  pyramid.  Equation  of  time. 
Aberration  of  light ;  its  discovery  by  Bradley, 
ix 


TABLE  OF  CONTENTS 

CHAPTER  PAGK 

VIII.     THE  CALENDAR 138 

History  of  the  calendar.  How  to  find  the  day  of  the 
week  for  any  date,  past  or  future.  Perpetual  calendars : 
how  to  make  and  use  them.  How  to  find  the  date  of 
Easter  Sunday  in  any  year. 

IX.    NAVIGATION .151 

How  ships  find  their  way  across  the  ocean.  Method 
used  before  the  days  of  chronometers. 

X.     MOONSHINE 160 

Source  of  the  moon's  light.  The  lunar  months,  sidereal 
and  synodic.  Phases  of  the  moon.  Phases  of  the  earth. 
Air  and  water  absent  on  the  moon.  Occultations.  Meas- 
urement of  the  distance  of  the  moon  from  the  earth.  Axial 
rotation.  Librations.  Determination  of  lunar  diameter, 
volume  and  weight.  Lunar  day.  Harvest  rnoon.  Sun 
and  moon  in  the  almanac.  Moon's  true  orbit.  Measure- 
ment of  the  height  of  lunar  mountains. 

XI.     THE  PLANETS 183 

Kepler  and  Newton.  Central  forces.  Ptolemaic  theory 
and  Copernican  theory.  Planetary  periods.  Bode's  law. 
Modern  orbit  work.  Elements  of  orbits.  Measuring  and 
weighing  planets.  Satellites.  Mechanical  stability  and 
perturbations  in  the  solar  system.  Conjunctions  and  oppo- 
sitions :  visibility  of  planets. 

XII.     THE  PLANETS  ONE  BY  ONE .217 

Each  planet's  characteristics  considered  separately.  Hab- 
itability  of  Mars. 

XIII.  THE  TIDES 251 

Explanation  of  tidal  phenomena  due  to  lunar  attraction 
modified  by  solar  attraction.  Effect  of  the  tides  on  the 
moon  itself. 

XIV.  THE  SOLAR  PARALLAX   . 260 

Distance  from  the  earth  to  the  sun.  Modern  investiga- 
tions :  minor  planet  method;  Eros;  transit  of  Venus; 
Halley's  method ;  indirect  methods. 


TABLE   OF  CONTENTS 

CHAPTER  PAGE 

XV.     ASTRONOMIC  INSTRUMENTS     .        .        .        .        .    /   .        .    272 

The  telescope  :  magnifying  power ;  cross-threads  and  mi- 
crometers. The  meridian  circle  and  chronograph.  The 
equatorial.  Photographic  telescopes.  The  spectroscope. 

XVI.     SUNSHINE.        .        .        .        /       .'       .        .        .        .        .     286 

Constitution  of  the  sun.  Sunspots.  Measuring  and 
weighing  the  sun.  Theories  as  to  durability  of  the  sun. 
Photosphere,  chromosphere  and  corona.  Axial  rotation  of 
the  sun. 

XVII.     ECLIPSES.        .        .     ..        .:      .        .        .        ^        .        .297 

Explanation  of  their  cause.  Eclipse  limits.  Umbra 
and  penumbra.  Annular  eclipses.  Prediction  of  eclipses 
by  means  of  the  Saros.  Transits  of  Mercury  and  Venus. 

XVIII.     COMETS 307 

The  coma,  nucleus  and  tail.  Size  and  mass.  Danger  of 
collision  with  the  earth.  Light  pressure  theory.  Comet 
hunting.  Naming  comets.  Their  orbits.  Families  and 
relationships  of  comets. 

XIX.     METEORS  AND  AEROLITES      .        .        .        .        .        .        .     315 

Shooting  stars.  Showers.  Radiant.  Cause  of  light. 
Fragments  of  comets.  Height  above  the  earth's  surface. 
Aerolites  :  their  chemical  composition. 

XX.     STARSHINE 322 

Magnitude  and  brilliancy  of  the  stars.  Variable  and 
temporary  stars.  Stellar  eclipses.  Star  distances :  meas- 
urement of  parallax ;  the  light-year.  Motions  of  the  fixed 
stars :  proper  motion  and  radial  velocity.  Stellar  chemis- 
try. The  sun's  own  motion  in  space  :  the  apex.  Shall 
we  reach  Vega?  Statistical  studies  of  the  universe.  Stel- 
lar distribution.  Kinetic  theory  of  stars.  Binary  stars. 
Clusters  and  nebulae.  The  galaxy. 

XXL     THE  UNIVERSE  ONCE  MORE 356 

Origin  of  the  universe.  Laplace's  nebular  hypothesis. 
Chamberlin's  planetesimal  hypothesis. 

APPENDIX •        .363 

Elementary  mathematical  explanations. 
xi 


LIST   OF  PLATES 


PLATE 

1.  Comet  c  1911,  discovered  by  Brooks         ....        Frontispiece 

TO   FACE   PAGE 

2.  Spiral  Nebula  .        .     . ;.        .....        .        .     ,  .        .        4 

3.  The  Moon  in  the  First  Quarter  Phase      .        .        .        .        .        .       17 

4.  The  Samrat  Yantra,  "  Prince  of  Dials,"  at  Jaipur  ....      84 

5.  Precessional  Motion  of  the  Pole 133 

6.  Full  Moon  and  Crescent  Moon 162 

7.  Lunar  Enlargement .     182 

8.  Mars  and  the  Crescent  Venus 225 

9.  Discovery  of  Planetoids    .......  234 

10.  Saturn 242 

11.  Saturn 244 

12.  The  Lick  Observatory,  Mt.  Hamilton,  Cal.      .        .  272 

13.  The  Lick  Telescope  .....        .  .        .        .279 

14.  The  Crossley  Reflector      .         .        .        .  .        .        .        .281 

15.  Lick  Spectroscope .     283 

16.  Various  Spectra 284 

17.  The  Sun 288 

18.  Great  Sunspot .290 

19.  The  Prominences       .         .         .         .         .         .         .         .         .         .     293 

20.  Total  Solar  Eclipse,  with  Corona .     295 

21.  The  Morehouse  Comet,  Nov.  18,  1908      .        .        .         .        .        .     307 

22.  Halley's  Comet          .         .         .         .        .        .        .         .        .'       .     311 

23.  Meteor  Trail .        .        .        .     318 

24.  The  Constellation  Serpentarius        .         .         .        .        ...        .323 

25.  Nova  Persei       .         .        .'.-.'„        .        .        .        ...     328 

26.  Spectra 337 

27.  A  Star  Cluster  in  Hercules  and  the  Double  Star  Krueger  60  .         .349 

28.  The  Pleiades .        .        .    -     .     351 

29.  The  North  America  Nebula      .        .        .        .        .       :.        ..354 

30.  Nebulae      .        ....        .        .        .        .        .        .        ...     359 

31.  The  Trifid  Nebula    .         .         .        .        .        .        .        .        .        .360 

32.  Title-page  of  Newton's  Principia      .        .        .        .        ...       .     399 

xiii 


ASTRONOMY 


CHAPTER  I 

THE    UNIVERSE 

IF  a  company  of  men  and  women  should  chance  to  be 
gathered  together  on  some  clear,  quiet  evening  under  the 
dome  of  the  sky,  and  if  there  should  happen  to  come  into 
that  company  one  known  to  possess  an  acquaintance  with 
the  facts  and  the  theories  of  astronomic  science,  —  if  these 
things  should  occur,  inevitably  there  would  descend  upon 
that  astronomer  a  shower  of  questions.  These  he  would 
answer  in  simple  language,  after  a  kindly  fashion  for  many 
centuries  the  habit  of  his  guild;  and,  as  he  passed  on,  he 
would  once  more  marvel,  as  he  had  done  many  times  before, 
at  the  changelessness  of  man's  desires.  For  these  ques- 
tions of  the  multitude,  welcome  ever  to  the  star-man,  to-day 
still  resemble  those  that  were  laid  of  old  as  problems  before 
his  predecessors  at  the  side  of  the  pyramids. 

Why,  for  instance,  does  the  moon  appear  at  times  as  a  full 
round  disk,  at  others  as  a  tiny  crescent?  Why  do  certain 
bright  stars  called  planets  seem  to  wander  about  among 
the  multitude  of  their  fellows  ?  Why  is  summer  hot  and 
winter  cold  ?  How  do  navigators  find  their  way  across  the 
trackless  ocean  by  observing  the  heavenly  bodies  ? 

Let  us  begin  by  attempting  to  set  forth  as  best  we  may 
the  answer  to  some  of  these  many  eternal  questions  from  the 
skies.  For  the  astronomer  is  not  always  present ;  and  even 
B  1 


ASTRONOMY 

if  he  were,  it  is  often  better  to  gather  our  information  in 
silence,  by  means  of  the  process  called  reading.  The  very 
name  Astronomy  tells  us  what  our  science  is.  Derived 
from  two  Greek  words,  Astronomy  means  "the  law  of  the 
stars.77  Where  does  the  law  of  the  stars  hold  sway? 
Throughout  all  space.  What  is  space  ?  Space  is  the  place 
where  astronomy  has  its  being.  When  does  astronomy 
enforce  its  laws  ?  Throughout  all  time.  What  is  time  ? 
Time  is  the  period  during  which  astronomy  has  its  being. 
Astronomy  needs  no  logical  definitions  of  space  and  time. 
They  belong  to  it ;  they  are  part  of  it. 

Somewhere,  then,  in  the  endless  void  of  space  our  universe 
is  suspended ;  the  visible  universe.  Is  that  visible  universe 
but  one  of  many  ?  Are  there  invisible  universes  without 
number  scattered  through  the  vastness  of  space  like  conti- 
nents in  an  endless  ocean?  The  human  mind  loses  itself 
in  speculations  such  as  this :  nor  do  such  speculations  here 
concern  us ;  for  astronomers  consider  only  the  ascertainable 
phenomena  of  the  universe  that  unfolds  itself  to  our  senses. 

There  is  in  existence  a  vast  quantity  of  matter  and  a  vast 
quantity  of  active  energy,  or  force.  It  is  not  necessary 
at  this  point  to  define  these  terms ;  but  we  should  remember 
that  according  to  accepted  theory  the  total  of  matter  and 
the  total  of  energy  in  the  universe  do  not  change.  Matter 
is  never  destroyed ;  and  the  accepted  law  of  the  conservation 
of  energy  tells  us  that  the  quantity  of  energy  in  the  universe 
is  likewise  constant  and  unvarying  in  amount.  None  ever 
disappears  out  of  existence.  But  both  matter  and  energy 
may  and  do  undergo  changes  in  form  and  appearance. 
Thus,  water  may  appear  as  steam  or  as  ice  ;  and  the  energy 
of  a  moving  body  may  be  transformed  into  heat,  light,  or 
electricity. 

2 


THE   UNIVERSE 

When  we  examine  this  visible  universe  of  ours  at  night 
with  the  unaided  eye,  we  "see  several  different  kinds  of  ob- 
jects :  nebulae,  or  small  luminous  clouds ;  star  clusters,  like 
the  famous  group  called  the  Pleiades ;  individual  stars ;  the 
moon;  and,  occasionally,  comets  or  meteors.  In  the  day 
we  see  the  sun;  sometimes  the  moon;  and  very  rarely 
indeed  a  particularly  bright  star  or  comet.  We  shall  give 
here  a  brief  outline  of  existing  knowledge  concerning  these 
various  celestial  objects,  leaving  a  detailed  description  of 
their  peculiarities  to  later  chapters.  They  are  all  com- 
posed of  matter ;  all,  if  in  motion,  move  in  accordance  with 
the  laws  of  mechanical  science  which  govern  the  operation 
of  energy ;  and  all,  if  they  change,  undergo  only  changes 
such  as  accord  with  the  laws  of  physics  and  chemistry. 

First,  then,  the  nebulae.  We  shall  begin  with  these 
because  they  probably  represent  the  form  in  which  matter 
shows  itself  to  us  in  its  most  primitive  stage  of  development. 
Only  one  or  two  can  be  seen  with  the  unaided  eye;  and 
these  only  on  very  clear  nights  when  the  moon  is  invis- 
ible. In  the  telescope  they  appear  as  patches  of  luminous 
cloud,  often  more  or  less  irregular  in  form.  They  were 
once  thought  to  be  simply  conglomerations  of  small  stars, 
so  close  to  each  other  that  the  optical  powers  of  existing 
telescopes  were  unable  to  separate  them  into  constituent 
units.  This  view  gained  in  probability  for  a  long  time, 
because,  as  the  power  of  telescopes  increased  with  the  in- 
crease of  skill  among  opticians,  astronomers  were  con- 
stantly resolving  new  nebulae,  as  they  used  to  call  it ;  sepa- 
rating them  into  simple  close  clusters  of  faint  stars. 

But  the  invention  of  an  instrument  called  the  spec- 
troscope, in  the  middle  of  the  nineteenth  century,  put  us 
in  possession  of  a  means,  previously  non-existent,  for  dis- 

3 


ASTRONOMY 

tinguishing  with  certainty  between  the  light  of  incandescent 
gases  and  that  derived  from  incandescent  or  luminous  matter 
in  the  liquid  or  solid  stage.  With  the  spectroscope  astrono- 
mers have  been  able  to  ascertain  that  there  are  many  nebulae 
in  a  truly  gaseous  condition;  that  probably  most  of  these 
objects  are  gaseous  bodies ;  that  they  could  not  be  resolved 
into  stars,  even  if  terrestrial  man  possessed  to-day  tele- 
scopes more  powerful  than  he  is  likely  ever  to  have  at  his 
command. 

According  to  many  modern  theorists,  we  may  take  the 
nebulae  to  be  matter  not  yet  fashioned  into  stars.  This 
means,  of  course,  that  certain  forces  are  at  work  in  the 
nebulae;  forces  of  irresistible  power,  slow  in  action,  as  all 
cosmic  changes  must  be  slow  when  measured  by  the  life  of 
human  generations ;  but  sure  in  action,  too,  with  that  in- 
finite sureness  which  belongs  in  celestial  spaces.  These 
forces  doubtless  produce  motions  of  vast  import  within  the 
body  of  the  nebula ;  heat  is  doubtless  engendered ;  conden- 
sations occur  at  certain  points ;  nuclei  are  formed ;  prob- 
ably, finally,  one  or  more  stars  take  the  place  of  these  nuclei ; 
and  so,  perhaps,  is  the  original  nebular  material  transformed 
into  stars  such  as  men  see  clustered  upon  the  sky  of  night. 

Certainly  the  force  of  gravitation  must  be  active.  Since 
the  time  of  Newton,  in  the  seventeenth  century,  it  has  been 
known  that  there  is  a  force  of  gravitation ;  that  under  the 
influence  of  that  force  every  particle  of  matter  in  the  uni- 
verse attracts  or  pulls  every  other  particle  of  matter ;  that 
the  combined  effect  is  always  motion  of  some  sort,  each 
particle  pursuing  in  space  some  determinate  path  or  orbit 
under  the  influence  of  gravitational  attraction  exerted  by 
all  the  particles. 

The  most  recent  observations  of  nebulae  have  brought  out 

4 


Photo  by  Keeler,  May  10,  1899.     Exposure,  4  hours. 

PLATE  2.     Spiral  Nebula. 


THE   UNIVERSE 

the  fact  that  they  are  extremely  numerous ;  probably  many 
hundreds  of  thousands  exist,  although  only  about  ten  thou- 
sand have  been  catalogued.  This  fact  is  of  importance; 
for  if  we  are  to  regard  the  stars  as  a  product  of  development 
or  evolution  from  the  nebulae,  we  should  expect  these 
gaseous  bodies  to  exist  in  numbers  comparable  with  the 
number  of  the  stars  themselves. 

But  of  even  greater  interest  is  another  recent  observation 
of  the  nebulae.  The  predominant  type  seems  to  be  spiral 
in  form;  a  species  of  central  hub,  carrying  two  attached 
curved  spires,  like  a  whirling  wheel  with  two  very  flexible 
spokes  but  no  rim.  There  can  be  little  doubt  that  these 
nebulae  are  subject  to  internal  motions,  probably  rapid  in 
themselves,  but  appearing  infinitely  slow  to  us  because  of 
the  almost  inconceivably  vast  distance  by  which  we  are 
sundered  from  them. 

According  to  the  foregoing  theory,  which  admits  the  exist- 
ence of  irregular  as  well  as  spiral  nebulae,  we  should  expect 
to  find  the  stars  in  groups,  a  certain  number  assembled 
comparatively  close  together  near  certain  former  nebulous 
regions  within  the  sidereal  universe.  And  this  is  precisely 
what  we  do  find.  Usually  the  number  of  stars  thus  belong- 
ing together  is  small ;  very  frequently  but  a  single  star  can 
be  detected  with  the  telescope.  But  many  of  the  constit- 
uent stars  of  a  group  may  be  too  faint  to  show  themselves 
on  account  of  their  distance ;  often  they  are  all  probably 
too  faint,  except  the  large  one  that  may  have  resulted  from 
the  former  hub  or  center  of  the  parent  nebula,  if  it  was  one 
of  true  spiral  form.  Furthermore,  gravitational  attractions 
and  orbital  motions  may  have  commenced  among  the  stars 
of  every  group  even  before  they  had  become  separate  bodies. 
While  they  were  still  numerous,  frequent  collisions  must 

5 


ASTRONOMY 

have  brought  about  the  coalescence  of  two  or  more  into  a 
single  larger  unit.  In  short,  we  have  here  an  outline  of  a 
fairly  consistent  explanation  to  carry  us  forward  from  the 
nebular  stage  to  that  of  stellar  development,  a  theory  that 
leads  us  to  expect  star-groups  ranging  all  the  way  from  a 
single  visible  luminous  object  to  a  detached  assemblage 
closely  packed  in  a  globular  cluster. 

And  the  stars  are  simply  suns ;  our  sun  is  a  star.  There 
can  be  no  doubt  that  the  stars  are  not  the  tiny  twinkling 
points  of  light  they  seem  to  be.  Their  apparent  lack  of 
size  or  volume  is  simply  a  result  of  the  distance  by  which 
they  are  separated  from  us ;  their  twinkling  comes  from 
undulations  or  other  irregularities  in  the  ocean  of  terrestrial 
atmosphere,  or  aair,"  through  which  we  are  compelled  to 
view  them.  We  know  the  stars  to  be  self-luminous,  masses 
of  glowing  incandescent  solids,  liquids,  or  gases.  We  know 
the  stars  to  be  composed  of  chemical  elements  practically 
identical  with  those  found  on  the  earth.  We  know  the  stars 
to  be  subject  to  the  law  of  gravitation ;  that  every  particle 
of  matter  in  each  one  of  them  is  endowed  with  that  mys- 
terious quality  postulated  by  Newton,  the  power  of  pulling 
all  other  matter  in  the  universe.  From  all  these  known 
facts,  and  reasoning  by  analogy,  we  are  led  to  believe  the 
stars  to  be  suns,  more  or  less  like  our  own  sun,  though  by 
no  means  necessarily  in  the  same  stage  of  cosmic  develop- 
ment. All  are  doubtless  cooling  gradually  and  steadily  by 
the  constant  radiation  of  heat  into  space ;  some  have 
probably  reached  temperature  conditions  similar  to  those 
existing  in  our  sun ;  and  there  may  very  probably  be  some 
that  are  attended  by  planets  like  our  earth. 

The  stars  are  classified  according  to  their  so-called  magni- 
tudes, by  which  astronomers  mean  simply  their  lucidity  or 

6 


THE   UNIVERSE 

brightness,  not  their  actual  dimensions ;  the  first-magnitude 
stars  are  the  brightest,  the  fifth-magnitude  stars  the  faintest 
usually  visible  to  the  unaided  eye  under  ordinary  conditions. 
There  are  in  all  about  sixteen  hundred  stars  of  the  first  five 
magnitudes ;  and  only  about  one  half  of  these  can  be  seen 
at  any  one  time,  when  the  sky  is  perfectly  cloudless,  because 
the  other  half  are  always  concealed  from  view  below  the 
horizon.  The  stars  are  also  divided  into  a  series  of  so-called 
constellations ;  very  irregular,  even  grotesque  imaginary 
figures  of  men,  animals,  and  other  objects,  placed  in  the  sky 
by  the  astronomers  of  old,  and  retained  there  in  a  somewhat 
simplified  form  by  the  moderns,  principally  on  account  of 
an  unwillingness  to  destroy  the  ancient  landmarks  of  this 
venerable  and  venerated  science. 

The  stars  so  far  described  are  called  fixed  stars,  which  means 
that  they  do  not  change  their  relative  positions  in  space ; 
that  any  two  of  them  now  close  together  have  been  thus  in 
proximity  from  the  beginning,  and  will  remain  so  to  the  end. 
But  modern  researches  have  brought  out  the  fact  that  these 
apparently  fixed  stars  are  not  really  fixed  absolutely.  They 
have  motions  in  space ;  these  motions  seem  to  us  extremely 
slow  and  minute  simply  because  the  stellar  distances  are  so 
vast.  For  at  a  sufficiently  great  distance,  even  large  and 
rapid  motions  will  necessarily  appear  reduced  and  retarded. 
And  it  is,  in  fact,  quite  inconsistent  with  what  we  know  of  the 
laws  governing  gravitational  attraction  to  suppose  any  par- 
ticle of  matter  in  the  universe  to  be  really  fixed  in  position 
absolutely.  Everything  must  move;  must  be  following 
some  duly  appointed  path,  ever  contrasting  the  intricate  com- 
plexity of  nature  with  the  wondrous  simplicity  of  nature's 
order  and  nature's  law.  Even  our  sun,  regarded  as  a  star, 
cannot  be  fixed  in  space,  but  must  be -moving  majestically 

7 


ASTRONOMY 

through  the  void,  drawing  with  it  our  attendant  earth,  and 
ourselves  upon  it. 

And  if  the  stars  are  incandescent  suns,  we  must  expect  to 
find,  and  we  do  find,  that  some  among  them  undergo  inter- 
nal changes  that  make  their  visible  brightness  vary.  In 
certain  cases  slowly,  in  others  more  rapidly,  their  luminosity 
waxes  and  wanes  with  a  more  or  less  periodical  regularity. 
Now  and  again,  rarely  and  at  long  intervals,  some  special 
catastrophe  takes  place ;  some  convulsion  of  nature,  whereby 
a  new  star  is  made  to  blaze  forth  into  view  where  previously 
had  been  only  darkness.  Possibly  we  witness  in  such  cases 
the  result  of  a  sudden  collision  in  space  between  two  ancient 
suns  previously  cooled  through  the  ages,  and  long  since 
bereft  of  luminosity  and  of  life.  The  stars  that  change  their 
brilliancy  are  called  variable  stars;  those  that  blaze  forth 
suddenly  are  "new  stars, "  or  novce. 

As  we  have  already  stated,  the  sky  contains  stellar  sys- 
tems other  than  those  involving  but  a  single  visible  object. 
Of  these  probably  the  most  interesting  are  the  double  stars, 
composed  of  two  individuals,  often  of  different  colors.  These 
double  stars  appear  but  single  to  the  unaided  eye;  only 
when  the  powers  of  a  telescope  of  some  size  are  brought  into 
play,  is  it  possible  to  resolve  them  into  their  component 
parts.  In  the  field  of  view  of  such  an  instrument  the  stars 
all  appear  as  brilliant  points  of  light,  occasionally  glittering 
and  sparkling,  but  the  glitter  and  sparkle  are  imperfections 
caused  by  terrestrial  atmospheric  effects,  and  by  the  impos- 
sibility of  constructing  telescope  lenses  whose  surfaces  are 
ground  to  the  right  theoretic  shape  with  absolute  exactness. 
In  other  words,  the  stars  appear  in  the  telescope  much  as 
they  do  to  the  eye :  only  when  the  star  is  a  double,  the 
telescope  often  shows  it  as  such,  while  the  eye  is  unable  to 

8 


THE   UNIVERSE 

see  between  the  two  components.  And  it  is  a  very  impres- 
sive sight,  when  we  turn  a  telescope  upon  one  of  these  double 
stars,  to  see  the  two  tiny  points  of  light  projected  on  the 
deep,  fathomless  background  of  the  night  sky,  and  to 
realize  that  the  speck  of  darkness  between  them  is  a  bit  of 
abysmal  space. 

Sometimes  the  close  proximity  of  the  components  of  a 
double  star  is  fortuitous  merely.  The  two  objects  may 
simply  appear  close  together  through  happening  to  lie  in 
almost  exactly  the  same  direction  from  us.  But  one  of  them 
may  in  reality  be  behind  the  other,  and  at  a  distance  from 
us  immeasurably  greater  than  the  first.  In  this  respect 
astronomic  observation  differs  from  the  viewing  of  ordinary 
objects  on  the  earth.  If,  for  instance,  we  should  happen  to 
notice  two  men,  both  standing  at  points  almost  exactly 
north  of  us,  but  one  ten  times  as  far  away  as  the  other,  we 
would  at  once  detect  a  difference  of  distance  from  the  fact 
that  the  distant  man  would  appear  much  smaller  than  the 
near  one.  But  in  the  case  of  the  stars,  which  we  see  as 
points  of  light  merely,  we  could  gather  no  such  information. 
Even  if  one  of  the  stars  should  be  brighter  than  the  other, 
this  extra  brilliancy  might  be  due  to  a  higher  intrinsic  light- 
giving  power,  and  in  no  sense  a  result  of  greater  proximity. 

When  two  stars  thus  appear  close  together,  though  in 
reality  separated  by  a  great  distance,  they  probably  have 
nothing  in  common,  and  are  of  lesser  interest.  But  in 
certain  cases  the  two  stars  will  appear  close  together  through 
really  being  near  each  other  in  space.  Then  they  must 
belong  to  a  single  system;  have  probably  originated  in  a 
single  nebula ;  true  twin  suns,  bound  one  to  the  other  and 
the  other  to  the  one ;  held  by  the  invisible,  intangible,  but 
indestructible  power  of  gravitational  attraction. 

9 


ASTRONOMY 

In  addition  to  these  fixed  stars,  whose  motions  were  un- 
known to  the  ancients ;  whose  motions  are  so  slow  that 
generations  of  men  must  come  and  go  before  they  can  reveal 
themselves  to  the  unaided  eye,  —  in  addition  to  these  fixed 
stars,  the  night  sky  contains  five  other  bright  stars  called  of 
old  the  planets,  from  the  Greek  word  TrXavrJTrjs,  the  wanderer. 
They  have  been  named  Mercury,  Venus,  Mars,  Jupiter,  and 
Saturn.  The  most  conspicuous  thing  about  them,  when 
viewed  without  a  telescope,  is  their  peculiar  and  rapid 
motion  among  the  fixed  stars.  They  can  be  seen  to  make 
an  entire  circuit  of  the  heavens,  traveling  apparently  among 
the  fixed  stars,  in  brief  periods  of  time  ranging  from  about 
a  year  to  about  thirty  years.  Of  course  we  now  know  the 
cause.  These  planets  are  not  properly  stars  at  all ;  they 
are  like  the  earth,  attendants  of  our  sun,  revolving  around 
the  sun  in  perfectly  definite  paths  or  orbits,  and  in  perfectly 
definite  periods  of  time.  Compared  with  the  fixed  stars, 
they  are  all  extremely  near  the  sun.  And  being  all  thus  com- 
paratively near  the  sun,  they  are  of  course  also  all  compara- 
tively near  each  other ;  and  our  earth  being  one  of  the  num- 
ber, they  are  all  comparatively  near  the  earth,  too.  But  we 
have  just  seen  that  the  extreme  apparent  slowness  of  stellar 
motion  is  really  only  a  result  of  the  extraordinarily  great 
distance  by  which  we  are  separated  from  the  stars ;  as  this 
immensity  of  distance  does  not  exist  in  the  case  of  the 
planets,  of  course  their  apparent  motions  must  and  do 
appear  to  us  comparatively  rapid. 

Their  apparent  motions  are  also  complex  in  a  high  degree. 
Two  of  them,  Mercury  and  Venus,  move  around  the  sun  in 
orbits  smaller  than  that  of  the  earth,  and  therefore  entirely 
within  the  earth's  orbit ;  the  other  three,  Mars,  Jupiter, 
and  Saturn,  are  exterior  to  the  earth.  Mercury  has  the 

10 


THE   UNIVERSE 

smallest  orbit  of  all.  It  is  always  actually  quite  close  to  the 
sun,  and  therefore  always  appears  near  the  sun  when  seen 
projected  on  the  sky.  Of  course,  it  cannot  be  seen  when  the 
sun  is  visible  on  account  of  the  overwhelming  luminosity  of 
the  sun  itself.  Therefore  we  can  observe  Mercury  occa- 
sionally only,  just  after  sunset,  near  the  point  of  the  horizon 
where  the  sun  has  disappeared ;  or  just  before  sunrise,  near 
the  point  of  the  horizon  where  the  sun  is  about  to  make  its 
appearance.  It  is  thus  always  seen  in  the  evening  or  morn- 
ing twilight,  and  was  called  of  old  the  evening  star  or  the 
morning  star.  The  same  is  true  of  the  planet  Venus,  which 
attains,  however,  a  much  greater  apparent  distance  from 
the  sun. 

The  exterior  planets,  Mars,  Jupiter,  and  Saturn,  may  be 
seen  at  certain  times  throughout  a  wide  range  of  space  on 
the  sky,  and  at  any  hour  of  the  night,  all  of  which  phe- 
nomena will  be  explained  in  detail  in  a  later  chapter.  Still 
other  planets  exist ;  but  they  are  mostly  too  faint  for  the 
unaided  eye;  they  have  been  discovered  telescopically  in 
modern  times.  All,  together  with  our  sun  itself,  are  proba- 
bly the  result  of  gradual  changes  in  a  parent  nebula. 

The  planets  are  unlike  the  stars  in  still  another  important 
particular.  We  have  seen  that  the  stars  are  self-luminous, 
incandescent;  the  planets  are  quite  different,  and  give 
out  no  light  of  their  own.  They  shine  only  by  reflected 
light  which  they  receive  from  the  sun.  The  light  goes  from 
the  sun  to  the  planet ;  illumines  it ;  and  then  we  see  the  planet 
by  solar  light,  just  as  we  see  objects  in  a  room  by  reflected 
solar  light,  which  we  call  daylight.  This  produces  a  rather 
curious  telescopic  planetary  phenomenon  called  phase,  a 
phenomenon  which  is  most  conspicuous  also  in  the  case 
of  our  moon.  The  planets  are  globular  in  shape,  and 

11 


ASTRONOMY 

therefore  only  one  hemisphere  can  be  illumined  by  the 
sun  at  any  one  time.  But  the  planet  does  not  usually 
happen  to  turn  its  illuminated  hemisphere  directly  towards 
the  earth.  Therefore  we  usually  see  only  a  part  of  the  bright 
hemisphere,  and  this  often  looks  more  or  less  like  what  is 
called  a  half-moon.  In  other  words,  we  always  see  a  hemi- 
sphere of  the  globular  planet,  but  it  is  not  the  same  hemi- 
sphere which  is  turned  toward  the  sun,  and  which  is  therefore 
bright.  If  the  hemisphere  we  see  and  the  bright  hemi- 
sphere are  mutually  exclusive,  we  see  a  dark  or  " new-moon" 
phase.  If  the  bright  hemisphere  and  the  one  we  see  over- 
lap, we  see  a  crescent,  half-moon,  or  other  phase,  as  the  case 
may  be.  Among  the  planets,  Mercury,  Venus,  and  Mars 
show  the  most  conspicuous  phase  phenomena. 

Sir  John  Herschel  has  given  a  good  illustration  of  dimen- 
sions in  our  solar  and  planetary  system.  Represent  the 
sun  by  a  globe  two  feet  in  diameter.  Then  Mercury  will 
be  a  grain  of  mustard  seed  on  a  circle  164  feet  in  diameter 
with  the  sun  near  its  center ;  Venus,  a  pea,  284  feet  distant ; 
the  earth,  also  a  pea,  430  feet  away;  Mars,  a  pin's  head, 
654  feet ;  Jupiter  and  Saturn,  oranges,  distant  respectively 
half  a  mile  and  four-fifths  of  a  mile.  The  nearest  fixed 
star,  on  the  same  scale,  would  be  distant  about  8000  miles, 
not  feet.  This  illustration  brings  out  clearly  the  compara- 
tively minute  dimensions  of  the  solar  system  in  relation  to 
the  vastness  of  stellar  distances. 

In  actual  appearance  the  planets  differ  greatly  in  the  tele- 
scope; and  they  differ  especially  from  the  fixed  stars. 
For  even  our  most  powerful  optical  apparatus  will  not 
suffice  to  magnify  the  latter  so  as  to  make  them  appear 
otherwise  than  as  minute  points  of  light.  Many  of  them 
doubtless  possess  globular  dimensions  greatly  exceeding  any- 

12 


THE   UNIVERSE 

thing  we  find  in  the  solar  system ;  but  the  vast  distances 
cause  these  dimensions  to  shrink  into  mere  nothingness, 
even  in  our  largest  telescopes. 

But  in  the  case  of  the  planets  these  great  distances  do 
not  exist,  and  therefore  the  telescope  shows  their  spherical 
size  in  the  plainest  possible  way.  But  the  planets  differ 
greatly  one  from  the  other.  Jupiter  shows  a  bright,  nearly 
round  disk,  crossed  by  a  few  dark  straight  lines  or  bands. 
It  is  accompanied,  in  small  telescopes,  with  four  satellites 
or  moons,  which  can  be  seen  to  revolve  around  the  planet. 
At  times  they  pass  behind  the  planet  and  disappear;  and 
again,  one  or  other  of  them  is  so  placed  that  the  planet  inter- 
poses between  it  and  the  sun.  Then,  too,  it  disappears; 
for  the  satellites  also  shine  by  reflected  solar  light ;  and,  of 
course,  they  receive  none  when  Jupiter  is  placed  between 
them  and  the  sun.  Finally,  at  certain  other  times  a  satellite 
may  pass  between  Jupiter  and  the  sun ;  and  then  it  can  be 
seen  to  cast  a  small  round  shadow  dot  on  the  bright  surface 
of  the  planet.  Such  phenomena  are  called  eclipses. 

Saturn  is  the  most  beautiful  of  the  planets,  viewed  with 
a  telescope  of  moderate  size.  It  has  a  number  of  moons  or 
satellites,  mostly  too  small  to  be  seen  in  a  glass  of  low  power ; 
but  its  most  conspicuous  feature  is  the  famous  ring  of  Saturn. 
This  is  a  flat  disk  surrounding  the  planet,  and,  in  the  words 
of  Huygens,  who  was  the  first  to  explain  it  correctly,  nowhere 
" sticking  to"  the  planet.  The  ring,  like  the  other  bodies  of 
our  system,  shines  by  reflected  solar  light ;  and  it  is  always 
distorted  in  appearance,  as  seen  from  the  earth,  into  a  flat- 
tened oval  or  ellipse,  like  a  cart-wheel  seen  nearly  edgewise. 
At  certain  times  we  actually  do  see  it  exactly  edgewise,  and 
then  it  appears,  of  course,  like  a  thin,  straight,  bright  line 
against  the  dark  sky  background.  And  when  the  ring 

13 


ASTRONOMY 

appears  opened  up  to  a  considerable  extent,  we  can  see  this 
dark  background  of  the  sky  by  looking  through  the  openings 
between  the  ring  and  the  ball  of  the  planet. 

Mars  and  Venus  show  us  plain  bright  disks  of  moderate 
size,  exhibiting  in  small  telescopes  little  or  no  detail  of  any 
kind  in  the  way  of  markings  or  bands.  Their  most  conspic- 
uous feature  is  the  phase,  which  is  much  more  marked  than 
it  is  in  the  case  of  Jupiter  and  Saturn,  whose  phase 
phenomena  are  practically  altogether  unnoticeable.  This 
follows,  of  course,  from  the  fact  that  the  quantity  of  visible 
phase  is  due  to  proximity ;  and  Mars  and  Venus,  being  the 
planets  nearest  to  our  earth,  must,  of  course,  show  more 
phase  than  the  distant  planets  Jupiter  and  Saturn. 

Mercury,  as  we  know,  is  seen  only  in  the  twilight,  showing 
in  the  telescope  a  small  disk  with  marked  phases. 

Comets  are  occasional  visitors  to  the  solar  system.  They 
come  presumably  from  outer  space  in  the  course  of  their 
orbital  motions  under  the  influence  of  gravitational  and 
perhaps  other  forces ;  remain  for  a  time  in  the  vicinity 
of  the  solar  system ;  are  consequently  visible  to  us ;  and 
finally  retire  again  into  the  depths  of  space  whence  they 
came.  When  bright  enough  to  be  observed  without  the  tele- 
scope, they  commonly  exhibit  to  our  view  a  brilliant  come- 
tary  "head,"  containing  a  central  condensation  or  nucleus 
surrounded  by  a  mass  of  tenuous  luminous  haze,  and  to  it 
often  attached  a  long  visible  streamer  or  tail,  in  olden  times 
dreaded  by  all  as  a  possible  harbinger  of  wars  and  pestilence. 

All  these  cometary  phenomena  are  well  seen  in  the 
photograph  reproduced  as  a  frontispiece  in  the  present 
volume.  The  tail  in  this  case  has  more  than  one  streamer ; 
and  its  length,  as  photographed,  is  about  11°,  or  nearly 
one-eighth  the  distance  from  the  horizon  to  the  zenith. 

14 


THE   UNIVERSE 

The  tail  actually  seen  by  astronomers  was  at  one  time 
twice  as  long.  The  little  curved  lines  on  the  photograph 
are  star-images.  We  should  of  course  expect  these  to  be 
round  dots  in  the  picture ;  but  in  photographs  of  this  kind 
they  are  almost  always  drawn  out  into  little  curves,  for  a 
very  simple  reason.  The  telescope  is  aimed  accurately  at 
the  comet  when  the  exposure  of  the  photographic  plate  is 
commenced,  and  it  is  kept  thus  pointed  at  the  comet  during 
the  whole  duration  of  the  exposure.  This  of  course  makes 
a  "moved  picture"  of  the  stars,  as  photographers  would 
call  it.  For  the  comet  will  "  wander  "  among  the  stars,  like 
a  planet,  in  consequence  of  its  orbital  motion  in  space ;  and 
if  the  telescope's  movement  upon  its  stand  is  adjusted  cor- 
rectly to  allow  for  the  comet's  motion,  the  photographic 
images  of  the  stars  must  suffer. 

The  earth,  considered  as  an  astronomic  body,  is  but  one 
of  the  smaller  planets;  yet  in  one  respect  it  is  the  most 
important  of  all,  since  it  is  the  one  upon  which  we  live. 
Astronomers  have  been  able  to  ascertain  many  facts  about 
the  earth,  which  we  shall  for  the  present  summarize  with 
the  utmost  brevity,  postponing  all  detailed  description  to  a 
later  chapter.  We  know,  first,  that  the  earth  rotates  once 
daily  on  an  axis ;  that  this  rotation  carries  us  around, 
too ;  that  in  consequence  of  it  the  sun,  stars,  and  other 
heavenly  bodies  seem  to  rise  in  the  east,  climb  upward 
in  the  sky,  and  finally  sink  down  again  and  set  in 
the  west.  We  also  know  that  our  earth,  like  the  other 
planets,  travels  around  the  sun  in  an  orbit ;  that  it  requires 
a  whole  year  to  complete  a  circuit  of  that  orbit ;  that  in 
consequence  of  the  daily  axial  rotation  and  the  yearly  orbital 
revolution,  we  experience  the  phenomena  of  night  and  day, 
summer  and  winter,  —  phenomena  to  be  explained  fully 

15 


ASTRONOMY 

later ;  finally,  we  know  from  actual  measures  made  upon  the 
surface  of  our  planet  that  the  earth  is  a  slightly  flattened 
globe  about  8000  miles  in  diameter. 

The  moon  is  the  only  satellite  of  our  earth,  and  by  far  the 
most  conspicuous  object  in  the  night  sky;  the  most  beauti- 
ful and  interesting  of  all  the  heavenly  bodies  when  observed 
through  small  telescopes ;  and  important  especially  as  being 
our  nearest  neighbor  in  the  whole  wide  domain  of  cosmic 
space.  Again  summarizing  existing  knowledge  as  briefly 
as  possible,  the  moon  is  now  thought  by  astronomers  to 
have  once  formed  a  part  of  the  earth ;  to  have  been  set 
free  in  some  very  distant  age  in  the  past  by  the  action  in  some 
way  of  gravitational  and  possibly  other  forces.  It  revolves 
around  the  earth  in  an  orbit  somewhat  similar  to  the  earth's 
own  annual  orbit  around  the  sun ;  completes  such  an  orbital 
revolution  in  about  twenty-seven  and  one-quarter  days ; 
and,  in  consequence  thereof,  appears  to  make  a  complete 
circuit  among  the  far  more  distant  fixed  stars  and  planets 
in  the  same  period,  traveling  around  from  a  position  of 
apparent  proximity  to  any  given  fixed  star  back  to  the 
same  star  again  in  the  twenty-seven  and  one-quarter  day 
period.  It  is  not  self-luminous  or  incandescent,  but  shines 
by  reflected  sunlight  like  the  planets ;  in  consequence  of  its 
nearness  to  the  earth,  it  exhibits  the  most  pronounced  phase 
phenomena,  varying  all  the  way  from  the  full-moon,  down 
through  the  half-moon  stage,  to  actual  invisibility  at  the 
time  of  new-moon.  It  is  about  240,000  miles  distant  from 
the  earth ;  is  about  2000  miles  in  diameter ;  and  the  gravi- 
tational attraction  of  its  mass  upon  the  waters  of  terrestrial 
oceans  gives  rise  to  the  ebb  and  flow  of  the  tides. 

The  physical  or  actual  appearance  of  the  moon  is  not 
unlike  that  of  the  earth.  The  surface,  as  seen  in  the  tele- 

16 


Photo  at  Lick  Observatory. 

PLATE  3.     The  Moon  in  the  First  Quarter  Phase. 


THE   UNIVERSE 

scope,  is  very  much  broken ;  there  are  several  mountain 
ranges,  and,  especially  prominent,  a  great  number  of  large 
craters,  apparently  of  volcanic  origin,  and  usually  having 
a  mountain  peak  in  the  center.  Extremely  conspicuous 
features  of  the  lunar  surface,  as  seen  in  small  telescopes,  are 
the  very  black  shadows  which  are  cast  on  the  surface  when 
sunlight  falls  obliquely  on  the  mountains  and  craters. 
There  is  no  air  or  other  atmosphere,  and  no  water ;  nor 
have  we  any  reliable  evidence  that  any  perceptible  changes 
have  taken  place  in  the  volcanic  surface  features  since  accu- 
rate records  of  telescopic  observation  were  begun  by  men. 

To  complete  this  preliminary  brief  outline  survey  of  our 
subject,  it  remains  to  add  a  few  words  about  the  sun,  the 
central  body  of  our  solar  system.  The  sun  is  our  source  of 
light  and  heat ;  without  it  life,  as  we  know  it,  would  be  im- 
possible on  our  earth.  It  is  about  ninety-three  million  miles 
distant  from  us,  and  nearly  a  million  miles  in  diameter; 
within  its  vast  bulk  might  be  placed  the  earth  and  moon, 
together  with  the  entire  lunar  orbit  in  which,  as  we  have  said, 
the  moon  revolves  around  the  earth  in  twenty-seven  and  one- 
quarter  days.  The  sun  turns  on  an  axis  in  a  period  of  about 
twenty-five  terrestrial  days ;  its  surface  is  usually  marked 
by  the  well-known  sun  spots,  visible  in  small  telescopes 
plainly,  and  first  seen  by  Galileo,  when  he  turned  upon  the 
sun  probably  the  first  telescope  ever  made.  These  spots 
are  now  known  to  have  periods  of  special  frequency.  Every 
eleven  years  they  occur  in  greater  numbers  than  usual; 
and  this  period  of  eleven  years  is  in  some  mysterious  way 
connected  with  the  known  frequency  periods  of  auroras 
and  magnetic  storms  on  our  earth.  The  bulk  and  mass  of 
the  sun  are  so  great  that  its  gravitational  attraction  far 
exceeds  that  of  all  the  planets  combined.  It  thus  becomes 
c  17 


ASTRONOMY 

the  gravitational  ruler  of  the  whole  solar  system ;  around  it 
all  the  planets  may  be  said  to  revolve  in  their  duly  appointed 
paths  or  orbits. 

It  is  hoped  that  the  foregoing  brief  summary  of  astronomic 
science  may  help  to  awaken  a  desire  in  the  reader  to  possess 
more  detailed  knowledge ;  and  this  we  shall  endeavor  to 
give  in  later  chapters ;  perhaps  we  may  be  permitted  to 
conclude  the  present  one  by  calling  attention  to  the  value  of 
astronomy  for  practical  purposes  as  well  as  for  mental  dis- 
cipline and  study.  It  is  often  said  that  astronomy  is  a 
somewhat  detached  subject;  of  interest  certainly,  but  hav- 
ing little  or  no  close  and  intimate  relation  to  the  everyday 
affairs  of  human  life.  But  in  reality  the  converse  is  the 
truth.  Probably  no  other  of  the  more  abstruse  sciences 
enters  so  directly  and  so  frequently  into  our  daily  affairs 
as  does  astronomy.  There  are  at  least  three  services  per- 
formed by  astronomy  that  are  essential,  and  without  which 
civilization,  as  we  know  it,  would  be  impossible.  These 
things  are :  first,  the  regulation  of  time ;  second,  the  exe- 
cution of  boundary  surveys  and  the  making  of  maps  and 
charts ;  third,  navigation. 

Few  persons  stop  to  think  when  they  enter  a  jeweler's 
shop  to  correct  their  watches  by  comparison  with  the  jeweler's 
" regulator,"  or  when  they  communicate  by  telephone  with 
a  central  telephone  station  to  ask  for  the  correct  time, 
that  both  the  jeweler  and  the  telephone  operator  must 
themselves  have  some  source  of  correct  time  by  which  to 
regulate  their  regulators.  This  source  of  correct  time  is 
the  astronomical  observatory.  The  standard  observatory 
clock  is  itself  but  a  fallible  piece  of  machinery  fabricated  by 
fallible  human  hands,  and  it  can  be  kept  right  only  by  con- 
stant comparisons,  made  on  every  clear  night,  with  the 

18 


THE   UNIVERSE 

unvarying  time  standards  provided  by  nature,  the  stars 
themselves  in  their  courses.  For  instance,  time  observations 
of  the  stars  are  made  regularly  and  nightly  in  the  United 
States  Naval  Observatory  at  Washington,  the  chief  official 
astronomic  station  of  the  United  States  government.  With 
these  observations  the  standard  clocks  in  the  clock  room  of 
the  observatory  are  corrected  and  timed ;  and  from  these 
standard  clocks  electric  signals  are  sent  out  daily  in  accord- 
ance with  a  pre-arranged  schedule  so  that  time-balls  can 
be  made  to  indicate  the  exact  instant  of  noon  to  the  people, 
and  jewelers  and  others  may  correct  their  regulators. 
Thus  is  every  citizen  in  touch  with  the  astronomic  observatory 
almost  daily  and  of  necessity,  although  he  does  not  generally 
realize  the  fact  until  it  is  brought  specially  to  his  attention. 

And  the  matter  of  mapping  and  charting  is  equally  depend- 
ent upon  astronomy.  Ordinary  small  surveys  of  farms  or 
towns  may  be  made  by  ordinary  surveyor's  instruments 
without  constantly  having  recourse  to  astronomers.  But 
of  what  value  would  be  a  map  of  an  entire  continent  unless 
the  customary  latitude  and  longitude  lines  were  inscribed 
upon  it  ?  And  these  essential  lines  cannot  be  so  inscribed 
without  astronomic  observations.  Such  observations  must 
necessarily  be  made  specially  for  the  purposes  of  each  survey, 
and  the  consequent  calculations  always  depend,  too,  upon  cer- 
tain prior  astronomic  data  contained  in  published  astronomic 
" tables"  or  printed  books,  themselves  in  turn  based  on  aver- 
age or  mean  results  obtained  in  the  great  observatories  of  the 
world  during  the  last  couple  of  centuries  by  steady  con- 
tinuous systematic  study  and  observation  of  the  stars. 

Even  more  important  than  continental  maps  for  the 
progress  of  civilization  are  the  coast  charts  published  by  the 
various  governments  of  the  maritime  nations.  These  also 

19 


ASTRONOMY 

require  very  precise  latitude  and  longitude  lines ;  and  here, 
as  before,  recourse  must  be  had  to  astronomic  observations 
and  accumulated  astronomic  results. 

Finally,  navigation  itself,  upon  the  open  sea,  could  not 
proceed  successfully  without  astronomy.  Those  of  our 
readers  who  have  crossed  the  ocean  in  a  magnificent  modern 
steamer  may  have  seen  at  times  the  captain  or  navigating 
officer  "take  the  sun,"  as  it  is  called,  with  a  sextant.  Pos- 
sibly they  have  thought  that  after  making  such  an  obser- 
vation the  navigator  could  read  on  the  face  of  the  sextant 
the  exact  position  of  the  ship  at  the  moment,  its  latitude  and 
its  longitude  on  the  earth,  as  ordinarily  understood  in 
geography.  But  such  is  by  no  means  the  fact.  Before 
they  can  be  made  to  yield  this  essential  information, 
sextant  observations  must  be  subjected  to  a  somewhat 
laborious  process  of  numerical  calculation,  or  " reduction," 
as  it  is  called.  This  is  an  astronomic  process ;  and  in  carry- 
ing it  to  completion  the  navigator  again  requires  certain 
printed  tables  of  a  purely  astronomic  character.  These 
are  contained  in  a  book  called  the  " nautical  almanac," 
which  is  published  annually  in  various  languages  by  the 
several  civilized  governments  of  the  world.  And  again,  as 
before,  for  the  preparation  of  such  nautical  almanacs,  these 
governments  must  maintain,  and  do  maintain,  astronomic 
computing  bureaus,  manned  by  astronomers,  and  employing 
in  their  calculations  once  more  the  published  results  obtained 
by  astronomers  of  the  past  in  the  various  great  fixed  obser- 
vatories. The  details  of  all  these  astronomic  activities 
must,  of  course,  be  postponed  to  later  chapters ;  but  it  is 
hoped  that  enough  has  been  said  here  to  remove  from  the 
reader's  mind  the  possible  notion  that  astronomy  is  of  little 
or  no  practical  utility  in  the  ordinary  affairs  of  men. 

20 


THE   UNIVERSE 

But  far  beyond  and  above  all  this,  the  study  of  astronomy 
possesses  a  value  peculiarly  its  own,  as  a  means  of  mental 
training.  On  account  of  venerable  age  and  consequent 
approximate  perfection  of  knowledge,  this  science  is  char- 
acterized especially  above  all  others  by  the  peculiar  intri- 
cacy of  the  elementary  problems  it  presents,  and  by  the 
unusual  exactness  of  which  their  solutions  admit.  Further- 
more, notwithstanding  the  importance  of  its  direct  practical 
applications,  which  have  been  mentioned,  the  study  of  astron- 
omy is  peculiarly  free  from  any  materialistic  tendency, — 
from  any  connection,  in  short,  with  utilitarian  motives. 
It  is  not  a  vocational  study,  giving  knowledge  which  can  be 
sold  for  money  by  the  young  college  graduate  upon  his  entry 
into  practical  affairs.  But  it  is  preeminently  a  study  which 
will  give  a  clearer  outlook  upon  the  universe  in  which  we 
pass  our  lives,  preeminently  one  that  will  make  that  universe 
seem  a  pleasanter  place  in  which  to  live.  So  that  if  a  certain 
portion  of  our  time  is  to  be  devoted  to  studies  that  are 
not  strictly  vocational,  astronomy  will  surely  be  found  a 
profitable  and  desirable  subject.  And  surely  also  there  is 
much  to  be  gained  in  our  choice  of  studies  from  the  selection 
of  such  as  are  likely  to  arouse  a  real  interest  in  the  student  ; 
to  arouse  that  desire  for  knowledge  which,  once  awakened, 
will  make  the  task  of  the  teacher  an  easy  one.  Here  again 
astronomy  holds  a  most  favorable  place.  That  which  has 
its  being  within  the  confines  of  a  single  drop  of  water  is  as 
wonderful  as  are  the  motions  within  a  planetary  or  sidereal 
system.  But  the  animalcules  within  that  drop  of  water, 
though  their  number  be  myriad,  can  never  stir  our  deepest 
interest,  for  they  are  without  that  strong  appeal  to  the  imag- 
ination, without  those  vast  distances  and  mighty  forces,  the 
materials  of  astronomic  study  alone. 

21 


CHAPTER  II 

THE    HEAVENS 

PROBABLY  the  best  method  of  approaching  the  study  of 
astronomy  is  to  begin  with  those  observations  and  problems 
that  do  not  require  the  use  of  any  instruments  whatever. 
These  problems  are  surely  the  earliest  problems,  since 
men  of  old  must  have  begun  to  discuss  the  mysterious 
events  they  could  see  about  them  in  the  universe  long  before 
they  had  invented  even  the  rudest  instruments  of  measure- 
ment. 

Astronomy  is  a  study  of  the  sky ;  and  the  first  thing  to  be 
noticed  in  a  study  of  the  sky  is  the  sky  itself.  To  us  it  ap- 
pears at  night  like  a  great,  round,  blue,  hollow  dome  within 
which  we  are  standing.  To  its  interior  surface  seem  to  be 
attached  the  apparently  numberless  bright  twinkling  points 
of  light  we  call  stars.  In  the  day  it  carries  only  the  sun, 
and  perhaps,  too,  the  moon  rather  faintly  visible ;  and  in 
the  intermediate  periods  which  we  call  twilight,  and  which 
occur  at  dawn  and  at  dusk,  we  can  see  perhaps  two  or  three 
dim  stars,  called  morning  and  evening  stars.  We  know 
that  these  morning  and  evening  stars  are  certain  of  the 
planets,  which,  as  we  have  already  seen,  are  members  of 
the  solar  system  like  our  earth,  circling  around  the  sun, 
each  in  its  proper  path  or  orbit. 

But  there  is  no  real  dome  of  the  sky  above  and  around  us ; 
it  is  simply  an  optical  illusion,  a  creation  of  our  own  imagina- 
tion. Nevertheless,  it  is  most  convenient  to  imagine  it  to 

22 


THE   HEAVENS 

be  real,  because  we  can  thus  fix  our  first  astronomical 
ideas  to  something  tangible ;  and  by  a  consideration  of  this 
round  dome  as  if  it  actually  existed,  we  shall  be  able  to 
clarify  and  to  solve  many  interesting  problems.  Granting, 
then,  that  there  is  such  a  dome  above  us,  we  have  no  reason 
to  imagine  it  other  than  perfectly  round.  Let  us  regard  it 
as  a  great  hollow  ball  or  sphere;  astronomers  have  given 
it  the  name  Celestial  Sphere. 

The  next  question  is  whether  this  celestial  sphere  is  the 
same  sphere  everywhere.  Is  the  celestial  sphere  surround- 
ing New  York  identical  with  that  surrounding  the  city 
of  Capetown,  South  Africa  ?  The  answer  is :  yes.  The 
sphere  is  the  same  sphere  everywhere.  Theoretically,  the 
center  of  the  sphere  is  at  the  center  of  the  earth ;  and  since 
the  diameter  of  the  earth  is  about  eight  thousand  miles,  an 
observer  on  the  earth's  surface  will  be  distant  about  four 
thousand  miles  from  the  true  center  of  the  sphere.  But 
such  a  distance  as  four  thousand  miles  is  literally  a  .mere 
nothing  compared  with  the  infinitely  vast  distance  of  the 
celestial  sphere.  The  whole  planet  earth  shrinks  into  a 
mere  dot  in  comparison.  It  makes  absolutely  no  difference 
whether  you  are  on  the  earth's  surface,  or  could  be  transferred 
to  its  center,  you  would  s.ee  identically  the  same  imaginary 
celestial  sphere.  The  stars  and  other  heavenly  bodies, 
wherever  they  may  be  situated  around  us  in  space,  seem  to 
be  projected  upon  that  distant  celestial  sphere,  and  attached 
to  its  interior  surface.  Even  if  you  could  make  a  sudden 
jump  of  about  ninety- three  million  miles  from  the  earth  to 
the  sun,  you  would  still  see  the  same  identical  sphere,  much 
too  far  away  to  be  affected  by  such  a  little  change  in  the 
observer's  position.  Not  only  the  earth,  but  its  entire 
orbit,  including  the  sun,  shrink  into  a  dot.  Astronomy  is 

23 


ASTRONOMY 


truly  a  science  of  vast  distances.  But  there  is  this  essential 
difference  between  the  distance  of  the  celestial  sphere 
and  all  other  distances  in  the  science.  The  far-ness  (if  we 
may  use  such  a  word)  of  this  imaginary  sky  sphere  is  in- 
finitely greater  than  any  other  actually  known  and  meas- 
ured by  men. 

The  accompanying  Fig.  1  is  intended  to  illustrate  this 
notion  of  the  celestial  sphere.     The  large  circle  is  supposed  to 

represent  the  sphere; 
only,  of  course,  its  size 
cannot  be  made  big 
enough ;  the  reader 
must  imagine  it  ex- 
tended to  infinity. 
The  dot  E  at  the  cen- 
ter of  the  big  circle  is 
the  earth;  the  reader 
and  the  author  are  sup- 
posed to  be  standing 
on  the  surface  of  that 
dot.  The  tiny  circle 
represents  the  earth's 

FIG.  1.    The  Celestial  Sphere. 

annual    path    around 

the  sun,  the  sun  itself  being  the  larger  dot  at  the  center  of 
the  tiny  circle.  The  crosses  represent  stars  scattered  through 
sidereal  space  at  all  sorts  of  distances  from  the  earth.  The 
lines  with  arrows  passing  through  the  crosses  indicate  the 
points  on  the  interior  surface  of  the  celestial  sphere  where 
the  stars  will  appear  to  be  projected,  and  where  they  will 
seem  to  be  attached  to  the  interior  or  supposedly  visible 
surface  of  the  sphere.  The  longest  arrow  indicates  the  point 
on  the  sphere  where  the  sun  will  appear  projected,  that  arrow 

24 


THE  HEAVENS 

being,  of  course,  merely  a  straight  line  passing  from  the  earth 
to  the  sun  and  thence  continued  outward  to  the  sphere. 
For  the  sun  will  also  appear  to  us  as  if  attached  to  the 
interior  surface  of  the  sphere,  like  the  stars,  at  the  point 
indicated  by  its  arrow.  This  elementary  notion,  that  the 
various  celestial  bodies  will  appear  to  be  located  on  the 
sphere  at  the  points  shown  by  their  arrows  is  an  important 
idea,  and  one  that  is  not  at  all  difficult  to  grasp.  We  must 
not  forget  that  the  arrows  are  all  supposed  to  be  infinitely 
long ;  even  the  solar  arrow  is  infinite,  although  the  sun  dot 
and  the  earth  dot  are  very  near  each  other,  cosmically 
speaking. 

Having  thus  fixed  our  ideas  as  to  the  celestial  sphere, 
we  must  next  study  it  in  its  relation  to  the  various  objects 
that  appear  projected  upon  it ;  and  the  first  important 
thing  to  consider  more  in  detail  is  the  position  of  the  sun  on 
the  sphere.  We  have  already  seen  that  the  earth  travels 
around  the  sun  once  in  a  year.  The  path  or  orbit  in  which 
the  earth  thus  travels  is  an  oval  or  ellipse;  but  for  the 
purpose  of  a  first  approximation  such  as  we  shall  here  con- 
sider, we  can  take  this  path  of  our  earth  to 
be  a  circle,  with  the  sun  at  its  center. 
Now  this  circular  orbit,  like  every  circle, 
must  lie  entirely  in  a  single  plane  or  flat 
surface.  The  accompanying  Fig.  2  shows 
this  circular  approximate  orbit  of  the  earth 
E  moving  around  the  sun  at  the  center  S  FlG-  2-  The  Earth's 
in  the  direction  shown  by  the  arrow.  The 
single  plane  or  flat  surface  in  which  the  entire  orbital  path 
lies  is  here  of  course  the  flat  plane  of  the  paper  on  which 
this  page  is  printed. 

It  is  evident  that  the  earth,  being  always  in  its  orbit, 

25 


ASTRONOMY 

must  likewise  always  be  situated  in  the  plane  of  the  paper. 
And  the  sun,  being  at  the  center  of  the  circular  orbit,  must 
also  be  in  the  same  plane.  From  these .  considerations  fol- 
lows the  important  preliminary  principle  that  earth  and  sun 
are  both  constantly  in  a  single  plane.  To  this  important 
fundamental  plane  has  been  given  the  name  Plane  of  the 
Ecliptic. 

The  plane  of  the  ecliptic  is  defined,  then,  as  the  plane  in 
which  are  situated  at  all  times  the  sun,  the  earth,  and  the 
earth's  orbit  around  the  sun.  Now  let  us  extend  our  ideas 
so  as  to  include  the  celestial  sphere  in  our  consideration  of 
the  earth's  orbit.  Imagine  the  orbital  plane,  but  not  the 
orbit,  extended  or  stretched  outward,  indefinitely,  farther 
and  farther,  approaching  gradually  an  infinite  bigness,  until 
at  last  it  meets  the  imaginary  celestial  sphere.  Evidently, 
it  will  cut  out  a  circle  on  the  celestial  sphere,  just  as  though 
one  were  to  slice  a  round  orange  with  a  flat  cut.  The  line 
in  which  the  rind  of  the  orange  would  be  severed  by  such  a 
cut  would  then  be  a  circular  line ;  and  so  also  must  the 
line  cut  out  on  the  celestial  sphere  by  the  ecliptic  plane  be  a 
circle.  The  fact  that  the  sphere  is  an  immense  globe  and 
the  orange  a  small  ball  here  makes  no  difference.  The 
principle  is  the  same. 

It  is  possible  to  draw  a  little  more  information  from  the 
analogy  of  the  orange.  Wherever  we  slice  the  orange,  we 
obtain  a  circle ;  but  if  it  was  sliced  through  the  center,  the 
orange  would  be  cut  in  two  equal  halves,  and  then  the 
circle  would  be  the  largest  circle  that  could  possibly  be 
drawn  around  the  rind  of  the  orange.  Applying  this  to  the 
case  of  the  celestial  sphere  cut  by  the  ecliptic  plane,  we  see 
at  once  that  here  also  the  sphere  is  cut  in  two  equal  halves. 
For  the  earth,  as  we  have  seen,  is  at  the  center  of  the  celes- 

26 


THE   HEAVENS 

tial  sphere ;  and  therefore  the  ecliptic  plane,  which  passes 
through  the  earth,  is  also  a  cut  or  slice  through  the  center 
of  the  celestial  sphere.  Consequently,  the  circle  cut  out 
on  the  celestial  sphere  by  the  ecliptic  plane  produced  to 
infinity  is  a  circle  as  large  as  can  possibly  be  drawn  on  the 
celestial  sphere,  and  it  divides  that  sphere  in  two  equal  halves. 
Such  a  circle  drawn  on  a  sphere,  dividing  it  into  halves,  is 
called  a  Great  Circle  of  the  sphere.  The  particular  great 
circle  of  the  celestial  sphere,  cut  out  by  the  plane  of  the 
ecliptic  produced  to  infinity,  is  called  simply  the  Ecliptic. 

The  ecliptic,  then,  is  defined  as  a  great  circle  of  the  celestial 
sphere  cut  out  by  the  plane  of  the  earth's  orbit  around  the 
sun,  produced  to  infinity.  It  would  be  a  convenience  if 
some  one  could  go  up  to  the  sky  and  mark  out  the  ecliptic 
circle  upon  it  with  a  big  paint-brush.  While  this  is  im- 
possible, it  is  perfectly  easy  to  mark  it  upon  a  celestial  globe ; 
and  the  reader  is  advised  to  examine  such  a  globe,  when  he 
will  surely  find  the  ecliptic  plainly  drawn  upon  it. 

The  important  peculiarity  of  the  ecliptic  circle  is  this : 
the  sun  must  always  at  all  times  appear  to  lie  in  that  circle. 
And  the  reason  is  quite  simple,  as  shown  again  in  Fig.  3. 
Here  we  have  once  more  drawn  a  large  circle  to  represent 
the  infinite  celestial  sphere ;  and  the  dot  which  should 
represent  the  combined  sun,  earth,  and  earth's  orbit  around 
the  sun  is  shown  at  the  center,  magnified  into  a  circle.  The 
observant  reader  will  notice,  upon  comparing  Figs.  1  and  3, 
that  in  the  former  figure  the  earth  occupies  the  center  of 
the  sphere,  whereas  in  Fig.  3  the  sun  is  at  the  center.  But 
the  figures  are  interchangeable,  as  we  already  know,  because 
of  our  having  assigned  infinite  size  to  the  celestial  sphere. 

In  Fig.  3  the  smaller  circle  represents  the  earth's  orbit 
around  the  sun,  E'  and  E"  being  two  positions  of  the  earth 

27 


ASTRONOMY 

in  its  orbit.  The  corresponding  apparent  positions  of  the 
sun,  as  projected  on  the  celestial  sphere,  are  shown  at  Sf 
and  S".  For,  as  we  already  know,  if  an  imaginary  line  be 
drawn  from  the  earth  to  the  sun,  we  must  necessarily  see 


FIG.  3.    The  Ecliptic  Circle. 

the  sun  from  the  earth  along  the  direction  of  that  imaginary 
line;  and  if  the  line  be  extended  outward  until  it  pierces 
the  celestial  sphere,  the  sun  will  appear  to  us  projected  on 
the  sphere  at  the  point  where  the  sphere  is  pierced  by  the  line. 
Now  this  sight  line  from  the  earth  to  the  sun  will  neces- 
sarily lie  entirely  in  the  plane  of  the  earth's  orbit,  for  in  that 
plane  both  the  earth  and  the  sun  are  at  all  times  situated. 
Consequently,  the  sight  line,  when  extended  to  pierce  the 
celestial  sphere,  must  necessarily  always  pierce  that  sphere 
somewhere  on  the  circle  cut  out  on  the  sphere  by  the  plane 

28 


THE  HEAVENS 

of  the  earth's  orbit  produced  outward  to  infinity.  But 
this  circle  is  the  ecliptic ;  and  thus  we  have  a  proof  that  the 
sun  must  always  appear  on  the  sky  projected  upon  the 
ecliptic  circle.  And  it  is  certainly  a  most  remarkable  thing 
that  it  should  thus  be  possible  to  draw  an  imaginary  circle 
on  the  sky  such  that  at  all  hours  of  the  day,  on  every  day 
of  the  year,  and  of  every  year,  when  we  look  at  the  sun,  it 
will  appear  to  be  situated  at  some  point  of  that  circle.  Yet 
it  all  follows  quite  simply  from  the  above  elementary  con- 
siderations concerning  our  earth's  orbital  motion  around 
the  sun.  And  it  is  furthermore  already  equally  evident 
that  as  the  earth  progresses  around  its  orbit,  as  shown  by 
the  curved  arrow,  the  sun  will  appear  to  progress  around  the 
ecliptic  circle  with  a  rate  of  motion  corresponding  to  the 
earth's  own  motion  in  its  orbit. 

Figure  3  also  gives  a  good  opportunity  to  explain  the 
meaning  of  the  terms  "angle"  and  " angular  distance," 
which  we  shall  have  frequent  occasion  to  use.  An  angle  is 
defined  as  the  difference  in  direction  between  two  lines. 
Thus,  if  we  consider  the  lines  SS'  and  SS"  in  Fig.  3,  the  angle 
between  them  is  indicated  by  the  combination  of  letters 
S'SS".  Every  angle  is  thus  indicated  by  a  combination  of 
three  letters ;  the  middle  letter  of  the  three  always  indicating 
the  point  of  the  angle,  or  its  so-called  "  vertex."  The  cor- 
responding angular  distance  on  the  celestial  sphere  between 
S'  and  S"  is  the  arc  S'S" ;  and  such  angular  distances  must 
of  course  be  measured  in  degrees.  In  Fig.  3  the  angular 
distance  S'S"  is  about  120°,  or  one-third  of  an  entire  circum- 
ference of  360°. 

These  facts  about  the  ecliptic  constitute  one  of  the  most 
important  discoveries  of  the  very  earliest  astronomers.  The 
hazy  records  of  extreme  antiquity  indicate  that  the  Chinese 

29 


ASTRONOMY 

knew  the  ecliptic  and  had  measured  its  position  on  the  sky 
as  early  as  1100  B.C.  The  early  Greek  astronomers  of  Alex- 
andria certainly  knew  of  it  ;  for  instance,  we  have  fairly 
reliable  records  showing  that  Eratosthenes  (276-196  B.C.) 
measured  its  position  quite  accurately. 

The  next  important  phenomenon  to  which  our  attention 
must  be  directed  results  from  still  another  motion  of  our 
earth ;  namely,  its  axial  rotation.  As  we  all  know,  the  earth 
turns  on  its  axis  once  daily ;  a  motion  which  is  quite  distinct 
from  its  orbital  revolution  around  the  sun.  Both  motions 
take  place  simultaneously,  the  earth  traveling  around  the 
sun  in  its  orbit  while  it  is  at  the  same  time  spinning  on  its 
axis,  much  as  a  couple  of  waltzing  dancers  move  from  end  to 
end  of  the  room  while  at  the  same  time  spinning  rapidly 
around  each  other. 

This  terrestrial  rotation  has  an  immediate  effect  upon  the 
celestial  sphere  and  all  the  heavenly  bodies  which  appear 
projected  upon  it.  For  the  astronomer,  being  fastened  to 
the  earth,  turns  around  with  it,  perforce.  And  as  the  earth 
turns,  with  the  astronomer  attached,  it  is  constantly  pre- 
senting him  to  a  new  part  of  the  celestial  sphere.  Just  so 
a  dancing  couple  face  every  point  of  the  compass  in  suc- 
cession, in  consequence  of  their  spinning  motion,  and  quite 
independent  of  the  fact  that  they  are  also  moving  about 
in  the  room  at  the  same  time. 

This  turning  of  the  astronomer  successively  toward  dif- 
ferent parts  of  the  celestial  sphere  makes  that  sphere  appear 
to  him  as  though  it  were  turning  around  the  earth  instead  of 
the  earth  turning  within  it,  precisely  as  a  railway  passenger 
sees  fields  and  trees  apparently  flying  past  his  train,  although 
he  knows  these  objects  are  really  fixed  in  position,  and  him- 
self in  rapid  motion. 

30 


THE  HEAVENS 

The  axial  rotation  of  the  earth  takes  place  from  west  to 
east;  and  the  consequent  seeming  rotation  of  the  celestial 
sphere  is  from  east  to  west.  Objects  projected  on  the 
sphere  partake  of  this  seeming  motion ;  the  sun,  the  moon, 
the  morning  and  evening  stars,  and  all  the  other  stars. 

This  is  the  cause  of  day  and  night ;  of  the  rising  and  setting 
of  all  heavenly  bodies,  including  the  sun.  As  the  earth 
rotates  from  west  to  east,  they  all  seem  to  revolve  in  the 
opposite  direction  daily,  rising  from  beneath  the  eastern 
horizon,  slowly  climbing  the  sky,  and  again  sinking  down 
to  set  in  the  west.  These  facts  are  quite  generally  known 
with  respect  to  the  sun  and  moon,  but  comparatively  few 
are  aware  that  the  stars  also  rise  and  set.  It  is  reported 
that  Sir  George  Airy,  a  recent  astronomer  royal  of  England, 
used  to  say  that  not  more  than  one  person  in  a  thousand 
knows  that  the  stars,  like  the  sun,  rise  and  set.  Most 
people  think  the  stars  are  always  the  same,  simply  a  uniform 
countless  assemblage  of  thickly  clustered  luminous  points. 

Having  thus  explained  the  earth's  rotation,  we  must  next 
consider  its  rotation  axis.  Our  planet  earth,  in  its  rotation, 
turns  about  an  imaginary  line  or  axis  passing  through  its 
center  and  meeting  the  earth's  surface  at  the  north  and  south 
poles  of  the  earth.  Now  imagine  for  a  moment  this  rotation 
axis  extended  outward  in  both  directions,  farther  and  farther, 
until  at  last  the  two  ends  pierce  the  celestial  sphere  itself. 
They  would,  of  course,  mark  out  on  the  sphere  two  points 
corresponding  exactly  to  the  two  terrestrial  poles.  These 
two  points  are  called  the  north  and  south  Poles  of  the 
heavens,  or  the  celestial  poles.  The  long  line  joining  them 
is  the  axis  of  the  celestial  sphere,  and  a  very  short  bit  near 
the  middle  of  the  line  is  the  terrestrial  rotation  axis.  Figure 
4  again  shows  the  celestial  sphere,  this  time  with  the  earth 

31 


ASTRONOMY 


at  the  center,  magnified  from  its  proper  size  of  a  mere  dot, 
so  as  to  exhibit  the  earth's  rotation  axis  and  its  prolongation, 

the  axis  of  the  celes- 
tial sphere.  N  and  S 
are  the  north  and 
south  poles  of  the 
earth;  NS  is  the 
terrestrial  rotation 
axis ;  and  its  pro- 
longation to  the  ce- 
lestial sphere  marks 
out  N'  and  S',  the 
north  and  south 
poles  of  the  celestial 
sphere. 

Now  since  the  ap- 
parent rotation  of  the 
celestial  sphere  is  merely  a  result  of  the  earth's  turning, 
and  since  the  latter  takes  place  around  the  axis,  so  also  the 
great  sphere's  seeming  turning  must  take  place  about  this 
same  axis.  In  other  words,  all  the  stars  must  seem  to  re- 
volve nightly  around  the  two  poles  of  the  heavens.  Stars 
very  near  the  poles  on  the  sky  will  seem  to  turn  in  little 
circles;  those  farther  from  the  pole  will  seem  to  turn  in 
larger  and  larger  concentric  circles. 

Figure  5  shows  a  few  of  these  circles  in  which  the  stars 
appear  to  revolve  nightly,  and  indicates  that  those  near 
the  two  poles  of  the  heavens  are  small.  As  we  go  farther 
from  the  poles  the  circles  become  larger,  until  at  last  we 
come  to  stars  halfway  between  the  two  celestial  poles, 
where  the  largest  of  all  the  circles  occurs.  The  circles  are 
of  course  all  parallel ;  and  they  are  concentric  in  the  sense 

32 


FIG.  4.    The  Celestial  Poles. 


THE   HEAVENS 


FIG.  5.    Diurnal  Circles. 


that  their  real  centers  all  lie  on  a  single  straight  line,  the  axis 
of  the  celestial  sphere.     The  stars,  as  they  appear  to  revolve 
in  the  circles,  of  course 
complete  a  revolution 
every  twenty-four 
hours,  since  the  axial 
rotation  of  the  earth 
within  the    sphere   is 
the  true  cause  of  the 
whole     phenomenon  ; 
and  this  axial  rotation 
occupies   exactly  one 
day     of    twenty-four 
hours.     And   because 
of  this   daily  period, 
the  circles  are  called 
Diurnal  Circles  of  the 
celestial  sphere.     Diurnal  circles  are  defined,  then,  as  par- 
allel circles  on  the  celestial  sphere  in  which  the  stars  com- 
plete their  daily  apparent  rotation  around  the  celestial  poles. 
We  have  just  seen  that  the  largest  of  all  the  diurnal 
circles  is  the  one  halfway  between  the  two  celestial  poles; 
and  it  is  a  particularly  important  one.     It  of  course  divides 
the  entire  celestial  sphere  in  two  halves,  which  are  called 
the  northern  and  southern  celestial  hemispheres,  and  this 
largest  diurnal  circle  is  itself  called  the  Celestial  Equator.     It 
corresponds  exactly  to  the  equator  on  the  earth,  which 
similarly  divides  our   planet  into  northern  and  southern 
hemispheres.     In  fact,  it  is  clear  that  as  the  terrestrial  and 
celestial  poles  correspond  exactly,   so  also  the  terrestrial 
and  celestial  equators  must  correspond  exactly.     And  it  is 
therefore  also  possible  to  define  the  celestial  equator  in  a 
D  33 


ASTRONOMY 

manner  quite  analogous  to  the  definition  of  the  other  impor- 
tant great  circle  of  the  celestial  sphere,  the  ecliptic.  For 
if,  as  in  the  case  of  the  ecliptic  plane,  we  imagine  the  plane 
of  the  earth's  equator  stretched  out  and  extended  until  it 
finally  reaches  the  celestial  sphere,  it  will  cut  out  a  great 
circle  on  the  sphere,  and  this  great  circle  is  the  celestial 
equator.  So  we  might  define  the  celestial  equator  as  a 
great  circle  on  the  celestial  sphere  cut  out  by  the  plane  of  the 
earth's  equator  produced  to  infinity,  and  this  definition 
is  equivalent  to  the  former  one,  which  describes  the  celestial 
equator  simply  as  the  largest  of  all  the  diurnal  circles. 

Having  thus  defined  the  celestial  poles  and  equator, 
it  is  easy  to  carry  analogy  a  little  farther,  and  inquire  what 
corresponds  on  the  sky  to  latitude  and  longitude  on  the 
earth.  The  reader  will  recall  from  geography  that  when  we 
desire  to  define  the  position  of  a  place  on  the  earth  we  do  so 
by  giving  its  latitude  and  longitude.  Terrestrial  latitude 
is  defined  as  the  angular  distance  of  a  place  north  or  south 
of  the  earth's  equator,  and  terrestrial  longitude  is  its  angular 
distance  east  or  west  from  some  so-called  " prime  meridian," 
such  as  that  of  Greenwich,  England. 

Exactly  analogous  methods  are  used  for  defining  a  star's 
place  on  the  sky,  or  the  location  of  the  point  where  it  ap- 
pears to  us  projected  on  the  celestial  sphere.  Unfortunately, 
the  terms  celestial  latitude  and  longitude  have  not  been 
used  for  this  purpose.  Instead  of  these  terms,  astronomers 
use  the  words  " declination "  and  "right-ascension"  ;  which 
bear  the  same  signification  with  respect  to  the  celestial 
equator  that  terrestrial  latitude  and  longitude  bear  to  the 
equator  on  our  earth. 

It  is  of  interest  to  consider  here  the  initial  point  from  which 
astronomers  reckon  right-ascensions ;  for  there  is  no  prime 

34 


THE   HEAVENS 


meridian  on  the  sky  like  that  of  Greenwich  on  the  earth. 
Instead,  astronomers  use  an  initial  point  on  the  celestial 
equator,  and  from  it  the  right-ascensions  of  all  celestial 
objects  are  counted.  This  point  is  called  the  Vernal  Equi- 
nox, and  its  location  will  be  understood  easily  from  the  fol- 
lowing considerations. 

We  have  so  far  defined  two  great  circles  of  the  celestial 
sphere,  each  dividing  the  sphere  in  two  halves.  They  are 
the  ecliptic  circle  and  the  celestial  equator.  Now  these 
two  circles,  as  shown 
in  Fig.  6,  must  inter- 
sect at  two  opposite 
points  on  the  sphere ; 
for  any  pair  of  great 
circles  on  any  sphere 
must  evidently  do 
this.  These  two  op- 
posite points  are 
called  equinoctial 
points;  one  is  the 
Vernal  Equinox,  the 
other  the  Autumnal 

FIG.  6.    Two  Great  Circles  intersecting  at  Opposite 

have  occasion  farther  Points  of  the  Sky. 

on  to  explain  the  im-          (After  Cassini'a  A*tronomie>  *• 78-  Pari3- 1740-> 
portance  of  these  two  points  a  little  more  in  detail ;  for  our 
present  purpose  we  need  merely  remember  that  the  vernal 
equinox  point  is  by  universal   convention  selected  as  the 
initial   point  for  measuring  all  right-ascensions.1 

At  the  risk  of  seeming  somewhat  tiresome,  we  must  still 
add  to  these  rather  prolix  preliminary  explanations  a  very 

1,  Appendix. 
35 


ASTRONOMY 

few  more  necessary  definitions.     For  there  is  still  another 
important  great  circle  on  the  celestial  sphere,  again  dividing 
it  into  a  pair  of  halves,  but  a  different  pair  from  the  two 
hemispheres   north   and    south   of    the    celestial    equator. 
This  important  great  circle  is  the  Horizon.     In  astronomy 
the  horizon  is  precisely  the  same  thing  as  the  horizon  in 
ordinary  life.     It  is  defined  accurately  as  a  great  circle  on 
the  celestial  sphere  cut  out  by  an  infinitely  extended  level 
plane  touching  the  earth  at  the  point  where  the  observer 
stands.1     Of  course,  in  the  interest  of  exactness,  we  should 
note  in  passing  that  the  same  horizon  circle  would  be  cut 
out  on  the  sky  by  a  plane  parallel  to  the  first,  but  passing 
through   the   earth's   center   beneath    the   observer's   feet. 
This  is,  of  course,  again  a  result  of  the  fact  that  the  earth's 
radius  of  four  thousand  miles,  by  which  distance  these  two 
planes  are  separated,  is  a  perfectly  negligible  quantity  in 
comparison  with  the  infinite  distance  of  the  celestial  sphere. 
Having  defined  the  horizon,  it  is  easy  to  add  two  other 
definitions,  both  of  which  refer  to  astronomical  terms  having 
also  the  same  signification  precisely  that  they  bear  in  ordi- 
nary English.     These  are  the  Zenith,  which  is  simply  the 
point  of  the  celestial  sphere  directly  overhead,  and  therefore 
exactly  90°  distant  from  every  part  of  the  horizon ;    and 
Altitude,  or  angular  elevation  above  the  horizon.     Altitude 
is  defined  accurately  thus  :  the  altitude  of  a  celestial  body  is 
its  angular  distance  (p.  29)  above  the  horizon.     The  altitude 
of  the  zenith  is  thus  evidently  90°. 

As  we  now  know  the  meaning  of  the  two  points  on  the 

celestial  sphere  called  the  celestial  north  pole  and  the  zenith, 

it  is  possible  to  define  next  the  Celestial  Meridian.     This  is 

a  great  circle  drawn  on  the  sphere  from  the  celestial  north  pole 

1  This  plane,  in  mathematical  language,  is  a  plane  tangent  to  the  earth. 

36 


THE   HEAVENS 


to  the  zenith,  and  thence  extended  completely  around  the 
sphere  until  it  returns  again  to  the  pole.  Very  simple  con- 
siderations show  that  the  celestial  meridian  must  pass 
through  the  north  and  south  points  of  the  horizon.1 

The  accompanying  Fig.  7,  representing  a  celestial  globe, 
may  make  the  foregoing  description  clearer.  The  circle 
HVO  is  generally 
made  of  wood,  and 
represents  the  celes- 
tial horizon.  HPZAO 
is  usually  made  of 
brass,  and  represents 
the  celestial  meridian, 
passing  through  the 
celestial  pole  P,  the 
zenith  Z,  the  north 
point  of  the  horizon 
H,  and  the  south  point 
of  the  horizon  O.  The 
circle  ASQ  is  the  celes- 
tial equator,  every- 
where 90°  distant  from 
the  pole  P.  The  cir- 
cle BC  is  a  diurnal  circle.  ZV  is  a  flexible  strip  of  brass 
marked  with  degrees  and  pivoted  at  Z.  It  can  be  turned 
to  any  part  of  the  horizon,  and,  by  means  of  the  degree 
divisions  marked  upon  it,  we  can  measure  the  altitude  or 
angular  elevation  of  any  star  above  the  horizon.  Some  of 
the  constellation  figures  (p.  7)  are  also  drawn  on  the  globe. 

Having  now  defined  the  principal  circles  and  points  upon 
the  celestial  sphere,  let  us  next  investigate  the  position  of  the 

1  For  additional  definitions  and  explanations,  see  Note  2,  Appendix. 

37 


FIG.  7.     The  Celestial  Globe. 
(From  Lalande's  Astronomic,  3  ed.,  Tome  1,  p.  74. 
1792.) 


Paris, 


ASTRONOMY 


north  pole  of  the  sphere  with  respect  to  our  horizon.     We 
shall  first  imagine  an  observer  standing  at  the  north  pole 

of  the  earth.  It  is 
evident  from  the  ac- 
companying Fig.  8 
that  such  an  ob- 
server would  see  the 
celestial  pole  directly 
H  overhead,  in  the 
zenith.  For  P  being 
the  observer's  posi- 
tion at  the  north  pole 
of  the  earth,  PS  will 
be  the  earth's  rota- 
tion axis,  passing 
through  the  two 
poles  of  the  earth. 
And  if  this  axis  is 
lengthened  out  to  an  infinite  size,  it  will  meet  the  celestial 
sphere  at  P',  the  north  pole  of  the  sphere,  which  will  clearly 
be  directly  overhead.  PR  is  a  level  plane  touching  the 
earth  where  the  observer  stands ;  consequently  H  is  a  point 
of  the  horizon,  in  accordance  with  our  definition  (p.  36). 
OH'  is  a  plane  passing  through  the  earth's  center  parallel  to 
the  level  plane  PR ;  and  the  points  R  and  R'  will  coincide  on 
the  celestial  sphere  because  the  distance  PO  is  absolutely  neg- 
ligible in  comparison  with  the  infinite  distance  of  the  sphere. 
These  considerations  show,  that  to  an  observer  at  the  pole  of 
the  earth  the  celestial  pole  will  be  at  the  zenith,  and  its  alti- 
tude, or  angular  elevation  above  the  horizon,  will  be  90°. 

To  an  observer  standing  at  the  terrestrial  equator  the 
position  of  the  pole  will  be  quite  different,  as  shown  in  Fig.  9. 

38 


FIG.  8. 


Observer  at  the  North  Pole ;  Celestial  Pole 
in  the  Zenith. 


THE   HEAVENS 


If  we  place  the  observer  on  the  earth  at  E,  and  call  E  a 
point  of  the  equator,  the  terrestrial  rotation  axis  will  be  at 
PS,  because  any 
point  on  the  terres- 
trial equator  must 
be  90°  distant  from 
the  terrestrial  poles. 
This  puts  the  celes- 
tial north  pole  at 
P',  which  coincides 
with  K,  a  point  on 
the  horizon  of  an  ob- 
server at  E.  It  fol- 
lows from  this  that 
if  we  go  to  the  equa- 
tor of  our  earth,  we 
will  there  see  the  ce- 
lestial pole  in  our 
horizon,  distant  90°  from  our  zenith  at  Z,  directly  overhead. 
Having  thus  ascertained  that  to  an  observer  at  the  pole 
of  the  earth  the  celestial  pole  appears  overhead,  and  to  one 
at  the  equator  in  the  horizon,  it  is  not  difficult  to  realize 
that  an  observer  traveling  from  the  pole  to  the  equator 
will  see  his  celestial  pole  gradually  seem  to  move  down 
from  his  zenith  to  his  horizon.  For  if  the  celestial  pole 
occupies  two  extreme  positions  in  the  zenith  and  horizon 
when  the  observer  is  in  two  extreme  terrestrial  positions 
at  the  pole  and  equator,  it  is  clear  that  as  the  observer 
occupies  successive  intermediate  terrestrial  positions,  the 
celestial  pole  will  seem  to  occupy  successive  positions  also, 
intermediate  between  the  zenith  and  horizon.  This  is 
the  reason  why  travelers  going  south,  and  noting  the  pole 

39 


FIG.  9.     Observer  at  the  Equator ;  Celestial  Pole  in 
the  Horizon. 


ASTRONOMY 


star  night  after  night,  see  that  star  gradually  sinking  lower 
in  the  sky;  and  if  they  continue  southward  quite  to  the 
equator,  they  see  the  pole  star  actually  disappearing  at  the 
horizon.  For  the  pole  star  is  so  placed  in  space  as  to  be 
projected  on  the  sky  very  near  the  imaginary  celestial  pole ; 
and  consequently  the  visible  pole  star  partakes  of  the  changes 
which  we  have  just  explained.1  In  fact,  the  altitude,  or 
angular  elevation  of  the  celestial  pole  above  the  horizon,  is 
everywhere  equal  to  the  observer's  terrestrial  latitude,  or 
angular  distance  from  the  terrestrial  equator. 

This  very  important  theorem  enables  us  at  once  to  study 
the  very  different  appearance  of  the  celestial  sphere  and  its 

diurnal  circles  as  seen 
from  different  places 
on  the  earth.  At  the 
equator,  where  the 
pole  is  in  the  horizon, 
the  celestial  sphere 
looks  like  Fig.  10, 
called  the  Right 
Sphere.  Here  the 
diurnal  circles  (p.  33) 
are  all  perpendicular 
to  the  horizon,  and  they  are  all  bisected  or  halved  by  the 
horizon.  Consequently,  as  the  celestial  bodies  perform  their 
daily  apparent  rotation  with  the  sphere,  in  consequence  of 
the  corresponding  daily  axial  rotation  of  the  earth  inside,  — 
their  diurnal  circles  being  all  halved  by  the  horizon,  —  all 
the  celestial  bodies  will  be  above  the  horizon  just  as  long 
as  they  are  below  it.  They  will  be  "up"  twelve  hours,  and 
"down"  (or  "set")  twelve  hours. 

1  Note  3,  Appendix. 
40 


FIG.  10.    The  Right  Sphere. 
(After  Long's  Astronomy,  Vol.  1,  p.  91.    Cambridge,  1742.) 


THE  HEAVENS 


Now  we  have  seen  (p.  29)  that  as  the  earth  travels  around 
the  sun  in  its  annual  orbit,  the  sun  seems  to  travel  around 
the  ecliptic  in  a  corresponding  manner.  But,  wherever  it 
may  be  projected  on  the  ecliptic,  it  must  always  be  on  a 
diurnal  circle ;  in  the  light  of  what  we  have  just  learned  about 
the  right  sphere,  this  diurnal  circle  must  be  halved  by  the 
horizon ;  therefore,  to  an  observer  at  the  equator,  the  sun 
will  be  above  the  horizon  twelve  hours  every  day  in  the 
year.  We  therefore  see  that  at  the  equator  day  and  night 
are  always  equal  throughout  the  whole  year. 

Quite  a  different  state  of  things  holds  at  the  pole,  where 
we  see  what  is  called  the  Parallel  Sphere,  as  indicated  in 
Fig.  11.  Here  the  ce- 
lestial pole  is  at  the 
zenith,  and  the  diur- 
nal circles  are  all  par- 
allel to  the  horizon. 
If  a  celestial  body  is 
above  the  horizon  at 
all,  its  entire  diurnal 
circle  is  above  the 
horizon ;  it  will  re- 
main "up"  twenty- 
four  hours  during  each  axial  rotation  of  the  earth.  The 
largest  diurnal  circle,  the  equator,  here  coincides  with  the 
horizon ;  to  an  observer  in  the  northern  hemisphere,  stars 
between  the  celestial  equator  and  the  north  pole  never 
set ;  those  between  the  equator  and  the  south  pole  never 
rise. 

How  would  these  facts  affect  the  sun,  which  is  always 
seen  in  the  ecliptic,  as  we  know  ?  We  also  know  that  the 
ecliptic  is  halved  or  bisected  by  the  celestial  equator  (p.  35). 

41 


FIG.  11.    The  Parallel  Sphere. 
(After  Long's  Astronomy.) 


ASTRONOMY 


Therefore,  during  half  the  year  the  sun  will  be  between  the 
equator  and  the  north  pole.  During  that  half-year  its 
successive  diurnal  circles  on  the  parallel  sphere  will  be 
entirely  above  the  horizon,  and  the  sun  will  not  set.  This 
explains  the  important  and  well-known  fact  that  at  the 
north  pole  the  sun  remains  above  the  horizon  six  months, 
and  day,  as  well  as  night,  is  six  months  long. 

To  observers  situated  on  the  earth  in  places  like  New 
York,  intermediate  between  the  pole  and  the  equator,  the 

sky  appears  in  the 
form  called  the  Ob- 
lique Sphere,  shown 
in  Fig.  12.  Here  the 
diurnal  circles  are 
neither  perpendicular 
to  the  horizon,  nor 
parallel  to  it.  Being 
parallel  to  each  other, 
they  all  make  the 
same  angle  with  the 
horizon,  an  angle  which  is  different  in  different  terrestrial 
latitudes. 

And  the  diurnal  circles  are  not  halved  by  the  horizon, 
either.  Each  such  circle  is  divided  by  the  horizon  in  two 
unequal  parts.  If  the  circle  is  between  the  celestial  equator 
and  the  north  celestial  pole,  as  B,  Fig.  12,  the  part  above  the 
horizon  is  the  longer.  If  the  circle  is  between  the  equator 
and  the  south  pole,  as  ^-the  part  below  the  horizon  is  the 
longer.  Thus  it  follows  that  stars  projected  on  the  sky 
between  the  equator  and  the  north  celestial  pole  are  above 
the  horizon  each  day  longer  than  they  are  below  it,  and 
vice  versa.  Only  stars  on  the  celestial  equator  itself  have  a 

42 


FIG.  12.    The  Oblique  Sphere. 
(After  Long's  Astronomy.) 


THE  HEAVENS 

halved  diurnal  circle,  and  are  above  and  below  the  horizon 
equal  twelve-hour  periods.  Some  of  the  diurnal  circles 
quite  near  the  north  pole,  as  A,  do  not  reach  the  horizon 
at  all.  Stars  projected  on  these  diurnal  circles  will  there- 
fore never  set;  and  stars  with  corresponding  diurnal  circles 
near  the  south  pole  will  never  rise.1  Observers  in  the  south- 
ern hemisphere  of  the  earth,  of  course,  have  these  conditions 
reversed. 

The  sun,  always  projected  on  the  ecliptic,  may  have  its 
diurnal  circle  divided  either  way.  We  have  seen  (Fig.  6, 
p.  35)  that  the  ecliptic  is  bisected  or  halved  by  the  equator. 
Consequently,  when  the  sun  is  seen  in  one  half  of  the  ecliptic, 
it  is  between  the  equator  and  the  pole,  and  therefore  above 
the  horizon  longer  than  below  it ;  and  when  it  is  seen  in  the 
other  half  of  the  ecliptic,  it  is  below  the  horizon  longer  than 
above  it.  In  the  one  case,  the  days  are  longer  than  the 
nights ;  in  the  other,  the  nights  are  longer  than  the  days. 
As  the  sun  is  seen  in  one  half  of  the  ecliptic  during  about 
half  of  each  year,  it  follows  that  during  half  of  each  year 
our  days  are  longer  than  our  nights  in  the  temperate  regions 
of  the  earth,  where  the  oblique  sphere  prevails.  Only  when 
the  sun  is  exactly  on  the  equator,  at  one  of  the  two  points 
where  it  is  intersected  by  the  ecliptic,  does  the  sun  have  a 
halved  diurnal  circle,  giving  us  equal  periods  of  light  and 
darkness,  —  equal  days  and  nights.  We  have  already  seen 
that  these  two  points  of  intersection  of  the  equator  and 
ecliptic  are  called  equinox  points.  We  now  know  the 
origin  of  the  name ;  when  the  sun  is  seen  projected  at  either 
of  these  points  of  the  ecliptic,  we  have  equal  days  and  nights. 
It  may  facilitate  the  comprehension  of  these  facts  if  the 
reader  will  again  examine  Fig.  7,  p.  37,  the  Celestial  Globe. 

1  Note  4,  Appendix. 
43 


ASTRONOMY 

From  the  above  elementary  considerations  follows  at 
once  a  preliminary  understanding  of  the  phenomenon  called 
the  Seasons.  For  it  is  clear  that  the  half-year  during  which 
our  days  are  longer  than  our  nights  will  be  a  summer  or 
hot  half-year,  since  we  obtain  our  heat  from  the  sun ;  and 
the  half-year  with  the  long  nights  will  be  a  cold  or  winter 
half-year.  Near  the  terrestrial  equator,  where  the  right 
sphere  gives  constantly  equal  days  and  nights,  there  must 
be,  and  is,  a  complete  uniformity  of  seasons.  Near  the 
pole,  with  its  parallel  sphere,  there  is  a  long  six  months' 
summer  day,  and  a  corresponding  winter  night.  But  the 
polar  summer  is  itself  cold,  because  even  in  summer  the  sun 
never  rises  to  a  great  altitude  above  the  polar  horizon. 


44 


CHAPTER  III 

HOW   TO    KNOW  THE    STARS 

ANY  ONE  beginning  the  study  of  astronomy  quite  naturally 
desires*  to  proceed  as  quickly  as  possible  from  the  reading 
of  books  about  the  stars  to  an  examination  of  the  stars 
themselves  in  the  sky.  And  in  a  first  preliminary  survey 
it  is  of  interest  to  learn  the  names  of  the  principal  stars  and 
constellations  as  they  have  been  handed  down  to  us  from 
olden  times.  It  is  not  at  all  difficult  to  acquire  this  knowl- 
edge, now  that  we  have  become  acquainted  (Chapter  II) 
with  the  celestial  sphere,  and  the  more  important  lines  and 
circles  which  astronomers  imagine  to  be  drawn  upon  that 
sphere. 

There  are  four  objects  that  often  puzzle  beginners,  when 
they  attempt  to  compare  star  maps  with  the  night  sky  for 
the  purpose  of  identifying  the  more  important  bright  stars. 
These  are  the  same  four  things  that  so  puzzled  the  ancients, 
the  four  bright  planets,  Venus,  Mars,  Jupiter,  and  Saturn. 
Mercury,  the  only  other  bright  planet,  is  rarely  seen ;  but 
one  or  more  of  the  above  four  is  almost  sure  to  be  conspicu- 
ous in  the  sky,  to  a  certain  extent  impairing  the  correctness 
of  our  star  maps. 

For  these  star  maps  do  not  show  the  planets ;  and  for  a 
very  simple  reason.  We  know  (p.  10)  that  the  planets  seem 
to  wander  among  the  fixed  stars ;  they  appear,  now  here, 
now  there,  at  very  widely  varying  points  of  the  sky.  On 
the  other  hand,  the  fixed  stars  (p.  7)  retain  relative  posi- 

45 


ASTRONOMY 

tions  practically  unchanging;  if,  for  instance,  any  three 
are  located  in  a  straight  line,  they  will  continue  to  lie  on  that 
straight  line  for  centuries,  so  far  as  observations  with  the 
unaided  eye  can  ascertain.  It  would  be  possible  to  make  a 
star  map  showing  both  stars  and  planets  as  they  appear 
on  any  given  date.  But  such  a  map  would  not  be  correct 
six  months  later;  while  it  would  still  show  the  fixed  stars 
in  their  proper  relative  positions,  the  planets  would  be 
wrongly  placed,  on  account  of  their  wanderings.  For  this 
reason,  astronomers  omit  the  planets  altogether  from  their 
star  charts ;  and  beginners  are  puzzled. 

For  the  beginner,  upon  looking  at  the  sky,  always  ob- 
serves the  planets  first  of  all,  because  they  appear  as  bright  as, 
or  brighter  than,  the  most  brilliant  fixed  stars  on  account  of 
their  proximity  to  the  earth.  For  instance,  the  beginner 
may  see  three  lucid  stars  forming  a  small  triangle,  with  the 
brightest  star  of  the  three  at  the  apex  of  the  triangle.  He 
at  once  looks  at  the  star  map,  to  identify  this  triangle. 
Finding  none  in  the  proper  place,  he  always  concludes  that 
he  has  misunderstood  the  printed  directions ;  packs  up  his 
books  and  lantern,  and  returns  indoors,  discouraged.  The 
beginner  in  astronomy  is  always  modest  as  to  his  abilities, 
and  blames  himself  if  the  universe  fails  to  fit  the  printed 
directions.  Nor  does  any  real  astronomer  ever  lose  this 
modest  characteristic  of  the  beginner;  for  he  who  has 
studied  this  science  most  deeply  is  ever  most  of  all  convinced 
that  he  is  still  a  beginner. 

Of  course  the  absence, of  the  triangle  from  the  star  map 
was  simply  due  to  the  extremely  brilliant  object  at  the 
apex  being  a  planet,  and  therefore  properly  absent  from  the 
map.  The  triangle  on  the  sky  appears  on  the  map  as  a 
simple  straight  line  with  but  two  stars  upon  it. 

46 


HOW   TO  KNOW   THE   STARS 

Therefore  the  beginner  should  first  of  all  learn  to  know  the 
planets,  so  that  he  can  eliminate  them  in  comparing  his  star 
map  with  the  sky.  And  it  is  fortunately  easy  to  become 
familiar  with  the  planets,  perhaps  even  easier  than  to  learn 
the  stars.  We  have  merely  to  take  advantage  of  the  planets' 
superior  brilliancy  in  order  to  identify  them.  The  best 
way  is  to  make  observations  in  the  dusk,  after  sunset,  before 
the  stars  begin  to  become  visible.  If  there  is  any  bright 
planet  above  the  horizon  at  that  time,  it  will  be  the  first 
to  show  itself  in  the  twilight ;  it  will  be  the  evening  star. 

But  this  priority  of  appearance  in  the  evening  is  not  nec- 
essarily a  sure  test  for  distinguishing  the  planets;  for  if 
no  planet  is  above  the  horizon  at  the  moment  of  sunset,  the 
first  object  seen  in  the  dusk  of  the  twilight  sky  will,  of  course, 
be  the  brightest  of  the  fixed  stars  then  above  the  horizon. 
Therefore  it  is  important  to  have  another  criterion  for 
identifying  the  planets.  It  is  a  fact  that  the  planets  always 
appear  projected  on  the  sky  rather  near  the  ecliptic  circle 
(p.  27).  Therefore,  if  we  could  locate  the  position  of  the 
ecliptic  circle  on  the  celestial  sphere,  we  should  have  addi- 
tional evidence  as  to  whether  the  evening  star  appearing 
first  is  really  a  planet.  If  a  planet,  it  must  be  near  the 
ecliptic  circle. 

The  following  method  will  enable  the  beginner  to  locate 
the  ecliptic  circle  approximately  on  the  sky.  One  point 
of  the  circle  is,  of  course,  determined  by  the  position  of  the 
sun,  which,  as  we  know  (p.  27),  is  always  seen  projected 
on  that  circle.  Consequently,  as  we  are  making  these  ob- 
servations in  the  evening  twilight,  it  follows  that  one  point 
of  the  ecliptic  is  near  that  point  of  the  horizon  where  the 
sun  has  just  set. 

If  we  can  now  locate  on  the  sky  one  other  point  of  the 

47 


ASTRONOMY 

ecliptic,  we  can  determine  roughly  the  location  of  the  entire 
ecliptic  circle ;  for  two  points  are  sufficient  to  locate  any 
great  circle  on  the  sky.  This  can  be  done  best  by  mak- 
ing use  of  the  celestial  meridian,  which  we  recall  as  a  great 
circle  drawn  on  the  sky  from  the  zenith  directly  overhead 
down  to  the  south  point  of  the  horizon  (p.  36).  The  point 
of  the  meridian  crossed  by  the  ecliptic  can  be  ascertained 
from  the  following  little  table,  which  gives  the  roughly 
approximate  altitude,  or  angular  elevation  above  the  hori- 
zon, of  that  point  on  the  meridian  which  is  crossed  by  the 
ecliptic. 

To  use  this  table,  it  is  merely  necessary  to  face  the  south 
point  of  the  horizon,  and  imagine  the  meridian  drawn  ver- 
tically upward  on  the  sky  from  that  point  to  the  zenith 
overhead.  Next  we  must  imagine  the  entire  distance  on 
the  meridian  from  horizon  to  zenith  divided  into  ninety 
equal  degrees  or  spaces.  Then  the  table  gives  us  for  various 
terrestrial  latitudes,  and  for  various  dates,  the  number  of 
degrees  between  the  horizon  and  the  point  of  the  meridian 
at  which  the  ecliptic  crosses  it.  To  facilitate  the  practical 
use  of  the  table  we  have  placed  in  it,  next  to  each  number  of 
degrees,  a  simple  fraction  which  will  perhaps  be  more  con- 
venient in  making  actual  observations.  Thus,  where  the 
table  gives  46°,  we  find  also  the  fraction  ^,  meaning  that  the 
ecliptic  crosses  the  meridian  approximately  halfway  up 
from  the  south  point  of  the  horizon  to  the  zenith.  The 
fraction  J  belongs  with  46°,  because  46  is  approximately 
half  of  90°,  the  total  angular  distance  from  horizon  to  zenith. 

For  example,  if  we  should  observe  at  New  York  (approxi- 
mate latitude  40°)  at  sunset  on  January  1,  we  would  imagine 
a  great  circle  drawn  around  the  sky  from  the  sunset  point  of 
the  horizon  to  a  point  on  the  meridian  halfway  between 

48 


HOW  TO   KNOW   THE   STARS 


TABLE  FOR  FINDING  THE  ECLIPTIC  AT  SUNSET 

Angular  Altitude  of  its  Intersection  with  the  Meridian  above  the  South 
Point  of  the  Horizon 


LATITUDE  30° 

LATITUDE  40° 

LATITUDE  50° 

January  1     .     . 

58°  f  un 

46°  \un 

32°  \l 

February  1   ;    '. 

73   U 

62   |« 

49    \l 

March  1  .     .     . 

82   |« 

71    U 

61    IS 

April  1     ... 

-83   tj 

73   |J 

63   IS 

May  1      ... 

78   f  J 

66   |{ 

54   |J 

June  1      .     .     . 

75   f? 

52   |? 

39   H 

Julyl       .     .;. 

52   |? 

40   *f 

26    *f 

August  1       .     . 

42    |f 

31    I? 

20   if 

September  1 

38   |f 

28   If 

18   if 

October  1      .     . 

37    |? 

27   fsM 

17    If 

November  1 

39   U 

28   |" 

18   *? 

December  1  .     . 

45   4? 

34   |« 

22   |« 

the  zenith  and  tftte  south  point  of  the  horizon.  This  imag- 
ined line  would  be  part  of  the  ecliptic;  extending  it  beyond 
the  meridian,  and  around  the  sky  to  the  eastern  horizon, 
would  give  us  the  remaining  visible  portion  of  the  ecliptic. 
And  any  object  suspected  of  being  a  bright  planet  would 
necessarily  be  found  very  near  this  ecliptic  circle.  If  the 
moon  should  chance  to  be  visible  at  sunset,  it  would  give  us 
an  additional  point  near  the  ecliptic;  for  the  moon  likewise 
always  appears  in  the  immediate  vicinity  of  that  circle. 

Actual  observations  of  this  kind  will  of  course  be  extended 
in  the  twilight  for  about  an  hour  after  sunset.  As  the  posi- 
tion of  the  ecliptic  changes  somewhat  during  that  hour, 
we  have  added  two  letters  to  each  number  in  the  table. 
One  of  the  letters  is  either  a  u  or  a  d,  and  shows  whether 
the  ecliptic  point  on  the  meridian  is  moving  up  or  down  at 
the  moment  of  sunset.  The  other  letter  is  either  an  n  or 
E  49 


ASTRONOMY 

an  s,  and  indicates  whether  the  ecliptic  point  on  the  horizon 
is  moving  north  or  south  at  the  moment  of  sunset.  Thus, 
an  hour  or  more  after  sunset  on^ January  1  at  New  York  (lati- 
tude 40°),  we  should  draw  the  ecliptic  a  little  to  the  north 
of  the  observed  sunset  point  in  the  horizon,  and  a  little 
above  the  46°  point  on  the  meridian. 

If  we  find  a  brilliant  object  in  the  dusk  in  this  way  on  the 
ecliptic,  we  may  still  further  test  its  planetary  character 
by  the  absence  of  twinkling,  for  planets  do  not  twinkle  as 
much  as  stars.  If  the  suspected  object  shines  quietly, 
serenely,  almost  without  scintillation,  we  may  be  tolerably 
sure  it  is  a  planet. 

Still  another  important  aid  is  at  the  service  of  the  begin- 
ner in  his  planetary  search,  —  the  ordinary  almanac.  This 
will  tell  him  what  " evening  stars"  or  planets  are  visible 
on  the  date  when  he  makes  his  observations ;  and  it  is 
certainly  a  great  help  to  know  in  advance  whether  any 
planets  are  to  be  in  sight.  The  almanac  will  also  inform  him 
as  to  the  names  of  the  planets  he  may  expect  to  see. 

But  even  without  an  almanac  it  is  generally  easy  to  dis- 
tinguish between  the  different  planets.  Mercury,  when 
visible,  always  appears  very  near  the  horizon,  close  to  the 
point  where  the  sun  has  set.  The  best  date  to  look  for  it 
may  be  found  by  adding  successive  periods  of  one  hundred 
and  sixteen  days  to  the  initial  date,  Nov.  2,  1913.  The 
planet  can  usually  be  seen  for  a  few  days  before  and  after 
the  dates  obtained  in  this  way,  if  the  horizon  is  unusually 
free  from  cloud  or  mist.  Conditions  are  .most  favorable 
when  the  computed  dates  occur  in  the  early  part  of  the 
year,  from  January  to  May.  And  in  these  months  espe- 
cially it  is  important  to  begin  looking  for  Mercury  at  least 
a  week  before  the  predicted  dates. 

50 


HOW  TO   KNOW  THE  STARS 

Venus  should  be  sought  after  sunset  on  the  ecliptic ;  its 
angular  distance  from  the  sun  is  never  more  than  47°  (about 
one-quarter  of  a  great  semicircle  of  the  sky) ;  and  it  may  be 
much  less.  In  looking  for  it,  about  an  hour  after  sunset, 
we  must  remember  that  in  an  hour  the  sun  will  have  moved 
a  considerable  distance  below  the  horizon ;  therefore, 
even  if  Venus  is  47°  distant  from  the  sun,  we  must  expect 
its  distance  from  the  sunset  point  of  the  horizon  to  be  con- 
siderably less.  An  initial  date  when  Venus  attains  its 
greatest  distance  from  the  sun  is  Feb.  12,  1913.  Subse- 
quent occurrences  of  the  same  phenomenon  may  be  expected 
at  intervals  of  584  days  thereafter  (1.60  years).  These 
dates  are,  of  course,  highly  favorable  for  observing  the 
planet.  Both  Mercury  and  Venus  are  extremely  bright. 

Mars,  Jupiter,  and  Saturn  also  always  appear  near  the 
ecliptic,  but  they  may  attain  very  great  angular  distances 
from  the  sun.  They  are,  in  fact,  directly  opposite  the  sun 
in  the  sky  at  certain  dates,  which  are  the  most  favorable 
dates  for  finding  these  planets.  The  dates  are : 

Mars,  Jan.  5,  1914,  and  thereafter  at  intervals  of  780  days 
Jupiter,  July  5,  1913,  and  thereafter  at  intervals  of  399  days 
Saturn,  Dec.  7,  1913,  and  thereafter  at  intervals  of  378  days 

When  thus  opposite  the  sun,  the  planets  are  easily  found. 
It  is  merely  necessary  to  imagine  a  straight  line  drawn  from 
the  sun  to  the  observer,  and  thence  continued  outward  to 
the  celestial  sphere  at  a  point  opposite  the  sun.  And  if 
we  imagine  the  line  drawn  an  hour  after  sunset,  we  must 
not  draw  it  from  the  sunset  point  of  the  horizon,  but  from 
the  sun  itself,  making  an  approximate  allowance  for  the 
sun's  having  moved  some  distance  below  the  horizon  during 
the  interval  of  an  hour  since  sunset.  On  these  critical  dates 

51 


ASTRONOMY 

Mars,  Jupiter,  and  Saturn  are  on  the  meridian,  due  south, 
at  midnight. 

For  some  time  after  the  critical  dates,  these  three  planets, 
always  remaining  near  the  ecliptic,  diminish  their  angular 
distances  from  the  sun  at  the  approximate  monthly  average 
rate  of  25°  for  Mars,  30°  for  Jupiter,  and  32°  for  Saturn. 
In  estimating  such  angular  distances  it  is  well  to  remember 
that  the  angular  diameter  of  the  full  moon  is  about  one-half 
a  degree.  Furthermore,  all  the  above  numbers  vary  some- 
what in  different  years.  The  interval  of  780  days  between 
successive  critical  dates  for  Mars  is  especially  variable :  it 
is  usually  only  about  750  days  when  the  predicted  date 
occurs  in  the  early  months  of  the  year. 

A  final  test  as  to  the  planets  may  be  obtained  if  the  ob- 
server has  a  small  telescope  or  good  field  glass  at  his  disposal. 
In  such  an  instrument  the  planets  show  their  round  disks 
quite  plainly,  while  the  fixed  stars  appear  in  the  field  of 
view  as  mere  points  of  light  without  any  visible  extension 
into  disks.  In  a  three-inch  telescope  Jupiter  shows  moons, 
usually  four,  and  Saturn  usually  exhibits  the  ring.  Most 
observers  detect  in  Mars  a  sort  of  reddish  or  ruddy  color. 

Coming  now  to  the  identification  of  the  fixed  stars,  we 
shall  employ  a  method  resembling  somewhat  our  procedure 
in  the  case  of  the  planets.  It  is  not  our  purpose  to  include 
in  the  present  volume  detailed  charts  showing  all  stars 
visible  to  the  unaided  eye,  but  rather  to  confine  our  atten- 
tion to  the  stars  of  especial  brilliance,  and  the  more  con- 
spicuous constellations  with  which  every  one  should  have 
an  acquaintance. 

The  first  things  to  find  in  the  sky  are  the  pole  star  and 
the  constellation  Ursa  Major  (Great  Bear  or  "  Dipper"). 
These  objects  are  near  the  north  celestial  pole,  and  very 

52 


HOW  TO  KNOW  THE   STARS 


far  from  the  ecliptic  ;  consequently,  the  planets  never  appear 
among  them  to  confuse  the  visible  configurations  of  stars. 
The  pole  star,  close  to  the  north  celestial  pole,  is  always 
elevated  above  our  horizon  by  an  angular  altitude  very 
nearly  equal  to  the  observer's  latitude  (p.  40).  To  find  it, 
we  must  therefore  face  the  north,  and  imagine  the  celestial 
meridian  drawn  on  the  sky  vertically  upward  from  the  north 
point  of  the  horizon  to  the  zenith.  The  pole  star  will  then 
be  found  almost  exactly  on  the  meridian,  and  elevated  above 
the  horizon  by  an  angle  equal  to  the  observer's  terrestrial 
latitude.  In  New 
York,  for  instance, 
it  will  be  elevated 
41°,  or  about  f  of  the 
total  angular  dis- 
tance from  horizon 
to  zenith.  The  pole 
star  is  not  very  bril- 
liant ;  being  of  the 
second  magnitude,  it 
will  be  inferior  to  sev- 
eral of  the  brightest 
stars  visible  in  vari- 
ous parts  of  the  sky. 
To  verify  this 
identification  of  the 
pole  star  we  make  use  of  Ursa  Major.  This  constellation  con- 
tains seven  stars,  not  of  the  first  magnitude,  arranged  as  shown 
in  Fig.  13.  This  figure  exhibits  the  constellation  as  it  appears 
in  the  sky  at  9  P.M.  about  April  21  in  each  year.  The  reader 
will  notice  that  the  two  end  stars  of  the  seven  are  in  the  me- 
ridian directly  above  the  pole  star,  and  that  they  point  almost 

53 


FIG.  13. 


The  Pole  Star  and  Ursa  Major  as  seen  at 
9  P.M.  on  April  21. 


ASTRONOMY 

exactly  toward  the  pole  star.  For  this  reason  these  two 
stars  are  called  "The  Pointers."  If  these  seven  stars  appear 
on  the  sky  occupying  the  position  shown  in  Fig.  13  with 
respect  to  the  pole  star  at  9  P.M.  about  April  21,  there  is 
no  doubt  that  the  pole  star  has  been  identified  correctly. 
In  using  Fig.  13,  the  reader  should  bear  in  mind  that  the 
constellation  Ursa  Major  will  appear  much  larger  on  the 
sky  than  it  does  in  the  figure.  The  scale  of  the  figure  has 
been  so  chosen  that  the  distance  of  Ursa  Major  from  the 
pole  star  is  proportioned  correctly  to  the  elevation  of  the 
pole  star  above  the  horizon ;.  and  this  choice  of  scale  makes 
the  constellation  appear  rather  small.  The  other  con- 
stellation figures,  14,  15,  17,  18,  19,  20,  21,  22,  are  all  drawn 
to  the  same  scale,  to  avoid  confusion ;  and  the  reader  must 
expect  all  these  constellations  to  be  larger  on  the  sky  than 
they  appear  in  the  figures. 

In  consequence  of  the  seeming  rotation  of  the  celestial 
sphere  about  the  pole  (p.  32),  the  pointers  will  further  occupy 
the  positions  shown  in  Fig.  14,  at  9  P.M.  on  the  several  dates 
indicated  in  the  figure. 

On  intermediate  dates  the  pointers  will  of  course  occupy 
intermediate  positions ;  and  with  the  help  of  these  figures 
the  reader  should  have  no  difficulty  in  finding  the  pole  star 
and  making  certain  of  its  identification  by  means  of  the 
pointers. 

There  is  one  other  interesting  constellation  near  the 
celestial  pole:  Cassiopeia,  the  "Lady  in  the  Chair."  It  is 
found  easily,  also,  by  the,  aid  of  the  pointers.  Imagine  a 
straight  line  drawn  from  the  pointers  to  the  pole  star,  and 
continued  beyond  the  pole  star  an  angular  distance  equal 
to  the  distance  between  the  pointers  and  the  pole  star. 
The  end  of  the  line  will  then  be  in  Cassiopeia,  and  the  appear- 

54 


HOW  TO  KNOW  THE  STARS 


55 


ASTRONOMY 

ance  of  that  constellation  is  shown  in  Fig.  15.  It  looks  like 
the  letter  W.  The  arrow  shown  in  the  figure  indicates  the 
direction  of  the  pole  star  from  Cassiopeia,  and  is  approxi- 
mately a  continuation  of  the  line  by  means  of  which  Cassio- 
peia was  found.  In  comparing  Fig.  15 
with  the  sky,  it  is  therefore  necessary  to 
turn  the  book  around  until  the  arrow  is 
nearly  parallel  to  the  direction  of  the 
pointers  from  the  pole  star.  This  would 
make  the  arrow  vertical  upwards,  as  shown 
in  Fig.  15,  at  9  P.M.  on  May  18,  and  verti- 
FIG.  15.  Cassiopeia  caj  downwards  at  9  P.M.  on  November  18. 
It  would  be  horizontal  to  the  right  on 
February  18,  at  9  P.M.  ;  and  horizontal  to  the  left  on  August 
18,  9  P.M.  On  intermediate  dates  the  arrow  would  of  course 
occupy  positions  intermediate  between  these  vertical  and 
horizontal  ones ;  always,  of  course,  at  the  hour  of  9  P.M. 

Having  thus  indicated  a  method  of  finding  the  two  impor- 
tant polar  constellations,  we  shall  next  show  how  to  identify 
the  brightest  fixed  stars  of  the  first  magnitude  visible  in  the 
United  States  and  Europe.  They  are  fifteen  in  number; 
in  the  following  list  we  have  arranged  them  in  the  order  of 
luminosity,  the  brightest  of  all  being  placed  first. 

To  find  these  stars,  we  shall  use  a  method  similar  to  that 
employed  for  locating  the  ecliptic  circle  on  the  sky.  Let 
the  observer  face  the  south  at  9  P.M.,  and  imagine  the  merid- 
ian drawn  on  the  sky  vertically  upward  from  the  south 
point  of  the  horizon  to  the  zenith,  directly  overhead.  Let 
him  once  more  imagine  the  meridian  divided  into  ninety 
degrees  or  spaces,  beginning  at  the  south  point  of  the  hori- 
zon, and  ending  at  the  zenith.  The  following  table  will 
then  tell  him  the  dates  on  which  the  various  stars  in  question 

56 


HOW  TO  KNOW   THE   STARS 

FIRST-MAGNITUDE  STARS 


NAME 

CONSTELLATION 

COLOR 

Sirius 

Canis  Major 

(Big  Dog) 

Blue-white 

Vega 

Lyra 

(Harp) 

Blue-white 

Arcturus 

Bootes 

(Bear-keeper) 

Orange 

Capella 

Auriga 

(Charioteer) 

Yellow 

Rigel 

Orion 

(Hunter) 

White 

Procyon 

Canis  Minor 

(Little  Dog) 

White 

Betelgeuse 

Orion 

(Hunter) 

Red 

Altair 

Aquila 

(Eagle) 

Yellow 

Aldebaran 

Taurus 

(Bull) 

Red 

Antares 

Scorpius 

(Scorpion) 

Red 

Pollux 

Gemini 

(Twins) 

Orange 

Spica 

Virgo 

(Virgin) 

White 

Fomalhaut 

Piscis  Australis 

(Southern  Fish) 

Orange 

Regulus 

Leo 

(Lion) 

White 

Deneb 

Cygnus 

(Swan) 

White 

appear  on  the  meridian,  and  their  altitude  or  angular  eleva- 
tion above  the  south  point  of  the  horizon  when  they  are 
thus  situated  on  the  meridian,  always  at  the  hour  of  9  P.M. 
The  date  of  reaching  the  meridian  at  9  P.M.  is  the  same  for 
all  terrestrial  latitudes ;  but  the  altitudes  vary  in  different 
latitudes,  and  are  therefore  given  in  the  table  for  latitudes 
30°,  40°,  and  50°.  If  the  observer's  latitude  is  intermediate 
between  30°  and  40°,  or  between  40°  and  50°,  he  can  of  course 
use  altitudes  intermediate  between  those  given  in  the  table. 
Sometimes  the  tabular  altitudes  are  a  little  greater  than  90°. 
This  indicates  that  the  stars  in  question  cross  the  meridian 
north  of  the  zenith.  To  see  them,  an  observer  facing  south 
would  need  to  bend  his  head  back  so  as  to  see  a  little  beyond 
his  zenith.  A  better  way  is  to  turn  around  and  face  the 
north,  when  the  stars  in  question  will  be  seen  very  near  the 
zenith. 

57 


ASTRONOMY 


The  identification  of  the  bright  stars  will,  of  course,  in- 
clude an  identification  of  the  important  constellations  in 
which  they  are  situated,  as  indicated  in  the  preceding  table. 

TABLE  TO  BE  USED  IN  FINDING  FIRST-MAGNITUDE  STARS  ON  THE 
MERIDIAN  AT  9  P.M. 


STAR 

DATE  ON  MERID- 
IAN, 9  P.M. 

ALTITUDE  ABOVE  SOUTH  POINT  OF  HORIZON 

Lat.  30  ° 

Lat.  40° 

Lat.  50° 

Sirius     .     .     . 

Feb.  15 

43° 

33° 

23° 

Vega      .     .     . 

Aug.  15 

99 

89 

79 

Arcturus    .     *  . 

June  10 

80 

70 

60 

Capella       .     . 

Jan.  23 

106 

96 

86 

Rigel     .     .     . 

Jan.  23 

52 

42 

32 

Procyon     .     . 

Mar.  1 

65 

55 

45 

Betelgeuse 

Feb.  2 

67 

57 

47 

Altair    .     ... 

Sept.  3 

69 

59 

49 

Aldebaran  . 

Jan.  13 

76 

66 

56 

Antares      .     . 

July  13 

34 

24 

14 

Pollux   .     .    , 

Mar.  2 

88 

78 

68 

Spica     .     .     . 

May  28 

49 

39 

29 

Fomalhaut 

Oct.  20 

30 

20 

10 

Regulus      .     . 

Apr.  8 

72 

62 

52 

Deneb   .     .  -  . 

Sept.  16 

105 

95 

85 

The  above  table  is  correct  at  8  P.M.  instead  of  9  P.M.  on 
dates  two  weeks  later  than  those  given  in  the  table.  It 
is  correct  at  10  P.M.  on  dates  two  weeks  earlier  than  the 
tabular  dates. 

To  facilitate  finding  the  bright  stars  on  dates  other  than 
those  on  which  they  reach  the  meridian  at  9  P.M.,  we  now 
give  another  table  containing  the  dates  when  these  stars 
rise  and  set  at  9  P.M.  as  seen  from  the  three  terrestrial  lati- 
tudes 30°,  40°,  and  50°.  In  addition  to  the  dates  of  rising 
and  setting,  the  table  contains  the  direction  (as  N.E.,  S.W., 

58 


HOW   TO  KNOW  THE  STARS 


NNW 


NNE 


WNW 


WSW 


etc.),  to  which  the  observer  must  turn  in  order  to  see  his 
star  rise  or  set.  In  making  these  observations  it  is  impor- 
tant to  remember 
that  the  immediate 
vicinity  of  the  ho- 
rizon is  usually  ob- 
structed by  trees, 
houses,  etc.,  and 
that  even  when 
these  obstructions 
are  absent,  the  ho- 
rizon itself  is  sel- 
dom entirely  free 
from  clouds  or  mist. 
Therefore  the  ob- 
server should  not 
expect  a  rising  star 
to  be  visible  for 
some  time  (possibly  as  much  as  an  hour)  after  9  P.M.  on  the 
tabular  date  of  rising;  and  he  may  expect  it  to  disappear 
from  view  some  time  before  9  P.M.  on  the  tabular  date  of 
setting. 

The  directions  N.W.,  S.E.,  etc.,  to  which  the  observer 
must  turn,  are  roughly  approximate  only ;  but  accurate 
enough  to  facilitate  finding  the  stars.  The  accompanying 
Fig.  16  shows  the  order  in  which  these  directions  follow  each 
other  around  the  horizon. 

The  table  on  the  next  page  is  correct  at  8  P.M.  instead  of 
9  P.M.  on  dates  two  weeks  later  than  those  given  in  the  table. 
It  is  correct  at  10  P.M.  on  dates  two  weeks  earlier  than  the 
tabular  dates. 

To  aid  still  further  in  the  identification  of  the  finest  con- 

59 


ssw 


FIG.  16.    The  "  Points  of  the  Compass." 


ASTRONOMY 


<M«M<NC^r-H(Mi-i 


llllll 


W   -1 
«-GG 


rd  •     fH     p       • 

I  ^      Is  - 

R    QJ  T!    r?  T!  -3 


fl    O         O 


60 


HOW  TO  KNOW  THE  STARS 


FIG.  17.    Auriga,  with  Capella. 


FIG.  18.    Cygnus,  with  Deneb. 


FiQ.  19.    Gemini,  with  Pollux. 


61 


ASTRONOMY 


FIG.  20.    Leo,  with  Regulus. 


FIG.  21.    Scorpius,  with  Antares. 


FIG.  22.    Orion,  with  Rigel  and  Betelgeuse. 


62 


HOW  TO  KNOW  THE   STARS 

stellations,  we  have  prepared  the  preceding  diagrams  exhibit- 
ing their  appearance  when  rising,  when  setting,  and  on  the 
meridian.  In  each  case  the  diagram  contains  an  arrow 
showing  the  direction  of  the  pole  star ;  and  the  dates  when 
the  several  constellations  may  be  seen  at  9  P.M.  can  be  taken 
from  the  preceding  tables. 

Those  of  our  readers  who  may  desire  to  extend  their 
knowledge  to  the  less  conspicuous  constellations  may  now 
do  so  easily.  It  is  merely  necessary  to  proceed  from  the 
constellations  already  known  to  those  not  yet  identified, 
by  the  aid  of  a  star  atlas.  In  doing  this  it  will  be  best  to 
look  for  the  known  constellations  and  first-magnitude  stars 
on  the  maps,  and  proceed  from  them  first  to  the  neighboring 
unknown  constellations.  There  is  little  difficulty  in  doing 
this ;  the  knowledge  of  a  few  stars  with  which  to  begin  is 
the  only  troublesome  part  of  the  problem.  It  is  hoped  that 
the  tables  and  diagrams  of  the  present  chapter  will  suffice 
to  remove  this  initial  difficulty. 

It  is  also  possible  to  identify  the  stars  by  means  of  a 
globe  such  as  that  illustrated  in  Fig.  7  (p.  37),  but  it  is  not 
easy  to  learn  the  method  of  using  a  globe  without  the  aid 
of  oral  teaching.  A  few  minutes'  explanation  from  some 
person  who  understands  the  use  of  the  instrument  is  better 
than  many  printed  pages  in  a  book.  There  is  also  another 
contrivance,  called  a  planisphere,  which  is  simple  in  use,  and 
much  less  costly  than  a  celestial  globe.  This  instrument 
represents  the  globe  projected  on  a  plane  or  flat  surface ; 
and  by  means  of  a  rotating  disk  of  cardboard,  it  shows  at  a 
glance  what  stars  are  visible  above  the  horizon  at  any  hour 
of  the  night  and  on  any  date  in  the  year.  Planispheres 
are  always  accompanied  with  printed  instructions  suitable 
for  use  by  a  beginner  in  astronomy. 

63 


ASTRONOMY 

In  a  study  of  the  present  chapter  the  reader  will  have 
noticed  that  we  have  given  practical  directions  for  finding 
the  stars,  without  elaborate  explanation  of  the  principles 
upon  which  these  directions  are  based.  This  will  enable 
him  to  commence  his  study  of  the  sky  without  waiting  until 
he  has  mastered  the  later  chapters  of  the  book ;  it  is  hoped 
to  increase  his  interest  by  thus  allowing  him  to  undertake 
practical  work  at  the  earliest  possible  moment. 


64 


CHAPTER  IV 

TIME 

WE  have  seen  (p.  19)  that  it  is  one  of  the  principal  duties 
of  the  astronomer  so  to  regulate  clocks  that  they  may  indi- 
cate accurate  time :  let  us  now  endeavor  to  explain  the 
meaning  of  the  word  "time"  in  astronomy.  We  shall 
make  use  of  our  definition  (p.  36)  of  the  celestial  meridian  as 
a  great  circle  of  the  celestial  sphere  passing  through 
the  celestial  pole,  the  zenith,  and  the  north  and  south 
points  of  the  horizon.  For  in  astronomy,  this  meridian 
plays  a  most  important  part  in  the  explanation  of  no  less  than 
four  different  kinds  of  time.  These  are  called  :  — 

1.  Sidereal  time.  2.  Apparent  solar  time. 

3.  Mean  solar  time.  4.  Standard  time. 

A  unit  of  some  sort  is  necessary  for  measuring  the  dura- 
tion of  these  various  varieties  of  time  :  and  for  this  purpose 
astronomers  use  the  Day;  though  not  the  same  "day"  for 
the  four  different  kinds  of  time.  There  is  a  sidereal  day, 
for  measuring  sidereal  time;  an  apparent  solar  day,  for 
apparent  solar  time;  and  a  mean  solar  day,  used  for  both 
mean  solar  and  standard  time. 

Let  us  consider  first  the  simplest  kind  of  time,  sidereal  or 
"star-time."  We  have  had  (p.  35)  a  definition  of  the  vernal 
equinox  as  one  of  the  points  on  the  celestial  sphere  at  which 
the  ecliptic  circle  crosses  the  celestial  equator ;  and  we  have 
already  made  some  use  of  this  important  point.  We  shall 
now  find  that  it  is  fundamental  also  in  the  measurement 
of  sidereal  time. 

F  65 


ASTRONOMY 

As  the  celestial  sphere  performs  its  diurnal  seeming 
rotation,  due  to  the  real  axial  turning  of  the  earth  within  it, 
the  vernal  equinox,  like  the  stars,  rotates  with  the  sphere.1 
Consequently,  once  during  each  complete  diurnal  rotation 
of  the  sphere,  the  vernal  equinox  will  cross  the  celestial 
meridian.  At  the  precise  instant  when  the  vernal  equinox 
thus  crosses  the  celestial  meridian,  the  sidereal  day  begins. 
As  the  seeming  turning  of  the  sphere  proceeds  from  east  to 
west,  the  vernal  equinox  will  begin  to  move  westward  from 
the  meridian  as  soon  as  the  sidereal  day  has  commenced; 
and  after  a  complete  rotation,  it  will  again  reach  the  meridian 
from  the  east.  The  sidereal  day  will  then  end,  and  at  the 
same  instant  a  new  sidereal  day  will  begin.  The  sidereal 
day  is  defined,  then,  as  the  interval  of  time  between  two 
successive  returns  of  the  vernal  equinox  to  the  meridian. 

The  sidereal  day  is  divided  into  twenty-four  sidereal 
hours;  and  these  hours  are  counted  continuously  from 
0  to  24,  without  using  the  letters  A.M.  and  P.M.  When  the 
vernal  equinox  is  exactly  on  the  meridian,  and  the  sidereal 
day  begins,  the  sidereal  time  is  Oh  Om  0s;  and  this  would  be 
the  time  indicated  on  the  dial  of  a  standard  sidereal  clock, 
if  the  clock  were  exactly  right.  Then,  after  the  vernal 
equinox  has  passed  the  meridian,  and  has  completed  one 
twenty-fourth  part  of  an  entire  diurnal  rotation,  it  is  lh  Om 
0s  sidereal  time ;  2h,  3h,  4h,  etc.,  follow  in  succession ; 
until,  at  23b  sidereal  time,  the  vernal  equinox  lacks  but  one 
hour  of  reaching  the  meridian  once  more. 

When  the  vernal  equinox  is  lh  west  of  the  meridian,  we 

say  that  its  "  hour-angle "  is  lh;    and  similarly  for  2h,  3h, 

etc.,  up  to  24h.     Thus  the  hour-angle  of  the  vernal  equinox 

at  any  moment  may  be  defined  as  the  quantity  of  rotation 

1  Cf.  Note  2,  Appendix. 


TIME 

made  by  the  celestial  sphere  since  the  vernal  equinox  was 
last  on  the  meridian,  this  rotation  being  measured  in  hours, 
minutes,  and  seconds,  and  an  entire  rotation  of  the  sphere 
corresponding  to  24  hours.  And  in  the  light  of  this  definition 
we  may  define  the  sidereal  time  at  any  instant  as  the  hour- 
angle  of  the  vernal  equinox  at  that  instant.1  Recurring  to 
our  definition  of  right-ascension  (p.  34),  it  may  be  here 
stated  as  an  additional  fact  that  the  right-ascension  of  any 
star  appearing  on  the  celestial  meridian  at  any  instant  is 
always  exactly  equal  to  the  sidereal  time  at  the  same  instant.2 

This  last  important  fact  calls  attention  to  a  simple  and 
interesting  relation  between  sidereal  or  star-time,  and  the 
stars  themselves.  If,  for  instance,  we  have  at  hand  a  correct 
sidereal  clock,  and  that  clock  indicates  3h  sidereal  time 
exactly,  then  any  star  whose  known  right-ascension  is  3h  may 
be  found  at  that  moment  on  the  meridian.  Furthermore, 
sidereal  time  enables  us  to  know  at  once  how  much  time  has 
elapsed  since  any  given  star  was  on  the  meridian.  Thus, 
at  4h  sidereal  time,  we  know  that  our  star,  whose  right- 
ascension  is  3h,  passed  the  meridian  one  hour  ago.  At 
5h  we  know  it  was  on  the  meridian  two  hours  ago,  etc. ; 
and  thus  we  know  approximately  where  to  look  for  it  in  the 
sky.3 

We  must  next  consider  the  explanation  of  solar  time,  and 
its  relation  to  sidereal  time.  Let  us  begin  with  apparent 
solar  time,  which  is  the  kind  of  time  kept  by  the  actual  sun, 
as  we  see  it  in  the  sky.  The  definitions  are  quite  similar 
to  those  we  have  already  given  for  sidereal  time.  The  unit  for 
measuring  the  duration  of  apparent  solar  time,  the  apparent 
solar  day,  is  defined  as  the  interval  between  two  successive 
returns  of  the  visible  sun  to  the  celestial  meridian.  The 

1  Note  5,  Appendix.        2  Note  6,  Appendix.          *  Note  6,  Appendix. 

67 


ASTRONOMY 

day  begins  when  the  sun  is  exactly  on  the  meridian ;  when 
the  axial  turning  of  the  sphere  has  carried  it  one  twenty- 
fourth  part  of  an  entire  diurnal  rotation  westward  from  the 
meridian,  astronomers  say  it  is  lh  apparent  solar  time,  etc. 
Following  the  analogy  of  sidereal  time,  we  may  define  the 
hour-angle  of  the  visible  sun  as  that  quantity  of  the  celestial 
sphere's  rotation  which  would  carry  the  sun  from  the 
meridian  to  its  actual  position  on  the  sky.  And  we  may 
then  define  the  apparent  solar  time  at  any  instant  as  the 
hour-angle  of  the  visible  sun  at  that  instant.  Astronomers 
do  not  use  A.M.  and  P.M.  :  apparent  solar  time  is  counted 
continuously  from  Oh  to  24h,  like  sidereal  time.1 

We  have  seen  that  successive  returns  of  the  sun  to  the 
meridian,  giving  the  solar  day,  and  successive  returns  of 
the  vernal  equinox,  giving  the  sidereal  day,  are  both  caused 
by  the  same  apparent  axial  rotation  of  the  celestial  sphere. 
We  are  therefore  confronted  by  the  question :  why  are  these 
two  kinds  of  day  not  exactly  equal  ?  To  answer  this  ques- 
tion, we  recall  (p.  27)  that  the  sun  appears  at  all  times 
somewhere  on  the  ecliptic  circle  in  the  sky ;  but  that  (p.  29) 
it  never  appears  at  the  same  point  of  that  circle  on  two 
successive  days. 

The  motion  of  our  earth,  in  its  annual  orbit  around  the 
sun,  makes  us  see  the  sun  projected  at  opposite  points  of 
the  ecliptic  circle  at  intervals  of  about  half  a  year.  Opposite 
points  of  the  ecliptic  circle  are  180°  apart ;  and  half  a  year 
contains  183  days.  Therefore,  the  sun  changes  its  apparent 
position  on  the  ecliptic  circle  about  180°  in  183  days,  or  one 
degree  daily.  Now,  to  simplify  matters,  let  us  imagine 
that  the  sun  appeared  at  the  vernal  equinox  exactly  at  noon 
on  a  certain  day.  We  already  know  that  the  sun  appears 

1  Note  7,  Appendix. 
68 


TIME 

at  the  vernal  equinox  once  each  year ;  let  us  now  imagine 
that  it  did  so  exactly  at  noon  on  one  of  the  days  in  some 
particular  year.  On  that  occasion,  the  apparent  solar  day 
and  the  sidereal  day  must  have  commenced  at  exactly  the 
same  instant.  For  the  one  kind  of  day  begins  when  the 
sun  is  on  the  meridian  ;  the  other,  when  the  vernal  equinox  is 
on  the  meridian.  On  the  occasion  when  they  were  both  on 
the  meridian  together,  both  days  must  have  commenced 
together. 

But  while  the  next  apparent  diurnal  rotation  of  the  sphere 
was  in  progress,  the  sun  did  not  remain  at  the  vernal  equinox. 
Its  daily  change  of  about  one  degree,  as  seen  projected  on 
the  ecliptic  circle,  must  have  made  it  appear  approximately 
one  degree  east  of  the  vernal  equinox  on  the  ecliptic,  by  the 
time  a  single  diurnal  rotation  had  been  completed.  There- 
fore, at  the  instant  when  the  vernal  equinox  again  reached 
the  meridian,  thus  completing  the  sidereal  day,  the 
sun  must  still  have  been  a  short  distance  east  of  the 
meridian.  The  diurnal  rotation  must  have  continued  a  little 
longer  to  bring  the  sun  to  the  meridian,  so  as  to  complete 
the  apparent  solar  day  as  well. 

From  these  considerations  it  follows  that  the  solar  day 
is  a  little  longer  than  the  sidereal  day.  The  difference  is 
about  four  minutes :  under  the  conditions  imagined  above, 
the  sun  would  have  reached  the  meridian  at  the  end  of  the 
day  about  four  minutes  behind  the  vernal  equinox.  At  the 
end  of  a  second  day  it  would  have  been  about  eight  minutes 
behind  the  equinox,  and  so  continuing  on  succeeding  days. 

Thus  there  is  a  constantly  increasing  difference  between 
solar  and  sidereal  time,  sidereal  time  gaining  about  four 
minutes  daily  on  solar  time.  If  a  solar  clock  and  a  sidereal 
clock  are  placed  side  by  side,  it  is  easy  to  follow  this  con- 


ASTRONOMY 

tinually  increasing  gain  of  sidereal  time  by  simply  making  a 
daily  comparison  between  the  two  clocks. 

It  is  evident  that  this  difference  of  the  two  clocks  will 
amount  to  24  hours  in  a  year,  since  4m  X  365  is  approxi- 
mately 1440  minutes,  or  24  hours.  And  the  actual  lag  of 
the  sun  is  a  little  less  than  4m,  just  enough  to  make  the  yearly 
gain  exactly  24  hours.  It  is,  in  fact,  evident  that  as  the 
sun's  apparent  motion  in  the  ecliptic  circle  is  due  to  the 
earth's  annual  orbital  motion  around  the  sun,  and  as  this 
orbital  motion  is  completed  in  a  year,  it  must  happen  at 
intervals  of  one  year  that  the  sun  must  return  again  to  the 
vernal  equinox,  and  everything  repeat  itself  once  more. 
The  sidereal  clock  will  gain  just  one  day  in  the  year ;  and  if 
it  agreed  with  the  solar  clock  at  the  beginning  of  the  year, 
the  two  clocks  must  again  be  together  at  the  end  of  the 
year.  Accordingly,  the  number  of  sidereal  days  in  the  year 
is  one  greater  than  the  number  of  solar  days.  And  the  whole 
difference  between  sidereal  and  solar  time  is  due  to  the  fact 
that  the  sidereal  day  depends  on  the  earth's  axial  rotation 
alone,  while  the  solar  day  depends  on  both  the  axial  rotation 
of  the  earth  and  the  daily  fraction  of  its  annual  orbital 
motion  around  the  sun. 

This  lagging  of  the  sun  behind  the  vernal  equinox  amounts 
to  4m  approximately  each  day,  as  we  have  seen,  but  this 
approximate  quantity  of  4m  is  itself  variable,  within  certain 
limits,  throughout  the  year.  The  reasons  for  this  variation 
will  be  explained  in  detail  in  a  later  chapter ;  but  one  reason 
is  quite  obvious.  The  earth  does  not  move  uniformly  in  its 
annual  orbit  around  the  sun.  And  since  the  sun's  apparent 
motion,  as  projected  on  the  ecliptic  circle,  is  simply  a  result 
of  the  earth's  orbital  motion,  it  follows  that  the  sun's  daily 
change  of  position  in  the  ecliptic  circle  is  not  uniform  either. 

70 


TIME 

Consequently,  the  lag  of  the  sun  behind  the  vernal  equinox 
will  not  be  the  same  each  day,  and  as  the  sidereal  days  are  all 
equal,  because  the  earth  rotates  uniformly  on  its  axis,  the 
solar  days  are  unequal. 

There  are  various  inconveniences  resulting  from  this  in- 
equality of  solar  days :  prominent  among  them  is  the  diffi- 
culty of  making  solar  clocks  that  will  run  with  other  than 
uniform  motion.  A  clock  keeping  pace  accurately  with  the 
inequalities  of  the  solar  day  would  be  almost  a  mechanical 
impossibility. 

Therefore  astronomers  have  adopted  an  imaginary  con- 
ventional mean  solar  time,  and  a  conventional  unit  for  it, 
the  mean  solar  day.  These  are  so  arranged  that  they  corre- 
spond accurately  to  the  average  performances  of  the  actual 
visible  sun  and  the  apparent  solar  day,  and  differ  as  little  as 
possible  from  them.  The  mean  solar  days  are  all  of  equal 
length.  We  can,  if  we  choose,  even  think  of  an  imaginary 
mean  sun  in  the  sky,  whose  hour-angle  from  the  meridian  at 
any  instant  will  be  the  mean  solar  time  at  that  instant.  Such 
a  mean  sun  would  occasionally  have  a  greater  hour-angle 
than  the  actual  visible  sun,  and  then  the  mean  solar  time 
would  be  later  than  the  apparent  solar  time.  The  mean 
solar  clock  would  be  fast  of  an  apparent  solar  clock,  if 
there  were  such  a  thing.  And  when  the  mean  sun's  hour- 
angle  was  less  than  that  of  the  visible  sun,  the  mean  solar 
clock  would  be  slow.  We  shall  return  later  to  the  difference 
between  these  two  kinds  of  solar  time  more  in  detail :  the 
above  explanation  is  sufficient  for  our  present  purpose. 

These  differences  between  mean  solar  time  and  apparent 
solar  time  are  never  greater  than  about  one-quarter  of  an  hour. 
But  the  difference  between  either  kind  of  solar  time  and 
sidereal  time  of  course  ranges  all  the  way  from  zero  up  to 

71 


ASTRONOMY 

24  hours.  It  is  zero,  as  we  have  seen,  when  sun  and  vernal 
equinox  are  together.  Then  solar  time  lags  behind  sidereal 
time  continuously  about  four  minutes  daily,  until  in  a  year 
the  accumulation  totals  one  day,  and  the  two  kinds  of  time 
are  together  again.  We  call  the  date  in  each  year  when  the 
two  kinds  of  time  agree,  March  21,  or  thereabouts.  This  is 
therefore  the  date  where  the  sun  appears  in  the  vernal 
equinox. 

These  facts  explain  clearly  the  varying  aspect  of  the  stellar 
heavens  night  after  night.  The  fixed  stars,  as  seen  pro- 
jected on  the  sky,  maintain  positions  practically  unchanging 
with  respect  to  the  vernal  equinox.  Any  fixed  star  will 
therefore  rise,  pass  the  meridian,  and  set  a  certain  definite 
number  of  hours  and  minutes  after  the  vernal  equinox. 
In  other  words,  it  will  do  these  things  every  night  at  the 
same  sidereal  time.  Consequently,  as  the  sidereal  time 
gains  about  four  minutes  daily  on  solar  time,  each  star  will 
rise,  pass  the  meridian,  and  set  about  four  minutes  earlier 
each  night  by  solar  time. 

For  instance,  referring  to  our  table  (p.  60),  we  find  that  at 
New  York  (approximate  latitude  40°)  Arcturus  rises  at 
9  P.M.  on  February  20.  On  February  21  it  will  therefore  rise 
at  8.56 ;  on  February  22,  at  8.52 ;  etc.  Two  weeks  after 
February  20,  Arcturus  will  rise  56  minutes  earlier,  or  approxi- 
mately one  hour.  This  explains  the  statement  (p.  59)  that 
all  the  stars  in  the  table  will  rise  at  8  P.M.  instead  of  9  P.M. 
two  weeks  after  the  dates  given  in  the  table. 

Having  now  explained  the  meaning  of  time,  it  becomes 
possible  to  set  forth  very  simply  the  astronomic  signification 
of  the  time  differences  existing  between  different  places  on 
the  earth.  Why  does  Chicago  time  differ  from  New  York  time 
or  London  time  ?  Recurring  to  our  definition  of  the  celestial 

72 


TIME 

meridian  (p.  36),  we  remember  that  it  passes  through  the 
zenith,  or  point  directly  overhead.  But  the  point  overhead 
in  London  does  not  coincide  with  the  point  directly  overhead 
in  New  York.  Therefore  London  and  New  York  will  have 
different  zeniths,  and  different  celestial  meridians. 

Furthermore,  we  have  just  explained  solar  and  sidereal 
time  to  be  the  hour-angles  of  the  sun  and  the  vernal  equinox 
from  the  celestial  meridian.  It  follows  that  if  London  and 
New  York  have  different  celestial  meridians,  all  hour-angles 
must  be  different  at  any  instant  in  the  two  cities.  Conse- 
quently, neither  sidereal  nor  solar  time  at  London  will  be  the 
same  as  New  York  sidereal  or  solar  time  at  the  same  moment. 
How  much  will  they  differ  ? 

To  answer  this  question  we  must  have  recourse  once  more 
to  geography.  The  reader  will  remember  that  the  surface 
of  the  earth  is  supposed  to  be  divided  by  a  series  of  lines  called 
terrestrial  meridians  of  longitude,  great  circles  drawn  on 
the  earth  from  the  north  to  the  south  terrestrial  pole.  We 
have  already  mentioned  (p.  34),  for  instance,  that  the  terres- 
trial meridian  of  Greenwich,  England,  is  the  prime  meridian 
for  reckoning  terrestrial  longitudes.  And  the  longitude  of 
New  York  is  simply  the  angle  at  the  north  pole  of  the  earth 
between  the  terrestrial  meridians  of  Greenwich  and  New 
York. 

Now  the  celestial  meridians  of  these  two  places  cor- 
respond on  the  sky  to  their  terrestrial  meridians  on  the 
earth.1  Therefore  the  angle  between  their  celestial  meridians 
at  the  north  celestial  pole  will  be  the  same  as  the  angle 
between  their  terrestrial  meridians  at  the  north  pole  of  the 
earth.  In  other  words,  it  will  be  the  same  as  their  terres- 
trial difference  of  longitude.  And  since  time  at  Greenwich 

1  Note  8,  Appendix. 
73 


ASTRONOMY 

or  New  York  is  simply  an  hour-angle  measured  from  the 
celestial  meridian  of  Greenwich  or  New  York,  it  follows 
that  the  difference  in  time  will  be  equal  to  the  longitude 
difference  of  these  two  places  on  the  earth. 

Many  beginners  grasp  this  matter  of  time  differences  more 
easily  in  another  way.  Because  the  sun  rises  in  the  east,  and 
moves  westward  in  the  sky,  and  because  New  York  is  west 
of  Greenwich,  the  sun  must  pass  the  celestial  meridian  over 
Greenwich  before  it  reaches  that  over  New  York.  There- 
fore, when  it  is  noon  in  New  York,  noon  has  already  occurred 
in  Greenwich,  and  it  is  already  afternoon  in  the  latter  place. 
Consequently,  Greenwich  time  is  later  than  New  York  time ; 
and  Greenwich  clocks  are  fast  of  New  York  clocks.  So  of 
any  two  places,  east  clocks  are  always  fast  clocks :  both 
words  end  in  ast. 

To  complete  this  part  of  our  subject  it  is  still  necessary 
to  explain  what  is  meant  by  standard  time,  the  ordinary 
time  in  actual  use  in  our  everyday  affairs.  It  has  no  direct 
connection  with  astronomy,  but  is  a  mere  conventional 
arrangement  designed  to  prevent  the  inconvenience  due 
to  the  fact  that  astronomical  mean  solar  time,  as  we  have 
seen,  is  practically  never  the  same  in  any  two  places  on  the 
earth.  It  is  not  possible  to  avoid  large  time  differences, 
such  as  exist,  for  instance,  between  Greenwich  and  New 
York.  But  there  is  no  reason  for  the  public  to  be  troubled 
with  minor  time  differences  of  a  few  minutes  only. 

The  plan  actually  adopted  is  as  follows :  Greenwich  is 
taken  as  the  initial  point  for  reckoning  all  standard  time. 
The  earth  is  then  divided  by  a  series  of  standard  meridians 
15°  or  lb  apart,  and  everywhere  the  time  of  the  nearest 
standard  meridian  is  adopted  arbitrarily  for  use  instead  of  the 
mean  solar  time  formerly  employed.  Thus  our  ordinary 

74 


TIME 

clocks  not  only  fail  to  conform  to  the  motions  of  the  actual 
visible  sun ;  they  no  longer  even  run  in  conformity  with  the 
imaginary  mean  sun.  But  the  standard  time  for  which 
they  are  regulated  differs  from  mean  solar  tune  by  a  constant 
difference  only  in  each  locality.  This  constant  difference 
is  the  tune  difference  already  explained,  as  it  exists  between 
the  terrestrial  meridian  of  the  locality  and  the  nearest  stand- 
ard  time  meridian. 

The  great  advantage  of  this  system  arises  from  the 
standard  meridians  having,  by  definition,  tune  differences 
that  are  exact  multiples  of  lh.  The  standard  tunes  of 
any  two  places  must  therefore  differ  by  an  exact  number 
of  hours,  without  minutes  or  seconds;  whereas  the  true 
mean  solar  time  difference  will  practically  always  be  an 
odd  fraction  of  hours,  minutes,  etc.  It  follows,  for  instance, 
that  a  traveler  going  from  New  York  to  Chicago  can  set  his 
watch  on  arrival  by  merely  turning  it  back  one  hour.  To 
make  his  watch  accord  with  Chicago  standard  time,  he 
does  not  need  to  consult  any  timepiece  in  Chicago.  If 
his  watch  was  correct  in  New  York  by  New  York  standard 
time,  it  will  be  similarly  correct  in  Chicago,  if  it  be  set  one 
hour  slow  of  New  York  time. 

We  shall  close  the  present  chapter  with  a  brief  explana- 
tion of  the  International  Date  Line.  This  is  another  con- 
ventional arrangement  intended  to  prevent  certain  difficulties 
arising  from  the  time  differences  that  confront  travelers 
who  circumnavigate  the  entire  earth.  A  person  going  east- 
ward from  Greenwich,  for  instance,  will  set  his  watch  one 
hour  faster  for  every  15°  of  longitude  he  traverses,  in  accord- 
ance with  the  explanations  we  have  already  considered. 

But  if  he  should  travel  entirely  around  the  earth,  and 
continue  the  same  treatment  of  his  watch,  he  would  find, 

75 


ASTRONOMY 

upon  his  return  to  Greenwich,  that  the  watch  had  been  set 
fast  a  total  of  24  hours  during  the  trip.  The  traveler  would 
apparently  have  gained  a  day ;  and  if  he  kept  a  daily  journal 
or  diary,  he  would  find  the  current  date  in  his  journal  one  day 
later  than  the  date  printed  in  the  London  morning  papers 
issued  on  the  day  of  his  return  to  Greenwich.  And  in  a 
similar  way,  if  the  traveler  had  proceeded  westward  from 
Greenwich,  his  diary  would  have  been  one  day  "slow"  of 
the  London  papers  on  his  return. 

Of  course  there  is  no  real  gain  or  loss  of  a  day.  If  the 
traveler  went  around  the  earth  with  uniform  velocity,  and 
made  the  circuit  in  24  days,  for  instance,  he  would  have 
changed  his  longitude  15°  daily,  since  15°  X  24  =  360°. 
This  would  make  his  daily  time  difference  just  one 
hour.  Therefore,  while  he  would  appear  to  gain  a  day  in 
24  days,  yet  each  of  these  24  days  would  be  only  23  hours  in 
length :  his  apparent  gain  of  one  day  would  be  offset 
exactly  by  his  loss  of  one  hour  on  each  of  24  consecutive 
days. 

The  above  inconsistency  is  not  convenient,  even  though  it 
is  apparent  merely,  not  real.  Therefore  it  has  been  agreed 
that  navigators  shall  change  their  date  arbitrarily  by  one 
day  when  circumnavigating  the  earth ;  and  that  they  shall 
make  this  change  when  they  reach  a  certain  longitude,  also 
arbitrarily  chosen  on  the  earth.  The  terrestrial  meridian 
of  longitude  thus  chosen  is  180°  distant  from  Greenwich. 
This  meridian  passes  through  the  Pacific  Ocean ;  it  is  most 
appropriate  for  the  purpose  because  comparatively  few 
ships  navigate  that  part  of  the  earth,  and  so  the  arbitrary 
change  need  be  made  but  rarely. 

But  it  has  not  been  found  possible  to  confine  this  change 
of  date  accurately  to  the  180°  meridian  of  longitude.  There 

76 


TIME 

are  certain  groups  of  islands  crossed  by  this  meridian,  and  it 
would  obviously  be  most  confusing  to  have  different  dates  in 
force  in  neighboring  islands  of  the  same  group.  Therefore 
an  arbitrary  irregular  line  has  been  drawn  on  the  map  of 
the  Pacific  Ocean,  and  called  the  international  date  line. 
Navigators  are  all  instructed  to  change  their  date  by  one  day 
when  crossing  this  line ;  skipping  a  date  if  they  are  proceeding 
westward,  and  counting  a  date  twice  if  they  are  moving 
eastward.  And  the  arbitrary  line  is  drawn  in  such  a  way  as 
to  avoid  as  far  as  possible  confusing  changes  of  date  in 
neighboring  islands  or  in  the  possessions  of  a  single  nation. 
It  may  be  remarked  also  that  some  of  the  ordinary  standard 
time  meridians  have  been  similarly  bent  a  little  at  certain 
points,  so  as  to  avoid  having  two  kinds  of  standard  time 
in  two  parts  of  a  single  city,  or  in  two  cities  very  near  each 
other. 


77 


CHAPTER  V 

THE   SUNDIAL 

BY  means  of  the  definitions  and  explanations  contained 
in  Chapter  IV,  we  can  now  solve  a  very  interesting  practical 
problem.  The  sundial  is  no  longer  an  instrument  of  essential 
importance  in  everyday  affairs,  since  time  is  now  universally 
measured  with  mechanical  clocks  and  watches;  but  it  still 
remains  a  most  instructive  toy,  and  is  as  much  as  ever  a 
desirable  ornamental  monument  in  gardens  and  other  public 
places. 

We  shall  confine  our  attention  to  one  of  the  simplest 
forms  of  the  instrument,  —  the  dial  drawn  on  a  horizontal  flat 
surface.  Upon  that  surface  is  erected  a  vertical  gnomon; 
and  the  shadow  of  this  gnomon  falling  on  the  dial  indicates 
the  hour  of  the  day  by  its  position  among  the  dial  lines. 
Our  problem  is  to  design  the  correct  shape  of  the  gnomon 
and  to  draw  the  lines  properly  upon  the  dial  itself. 

In  Fig.  23  we  give  a  sketch  of  a  complete  horizontal 
sundial.  The  gnomon  abc  is  made  of  a  piece  of  flat  brass 
plate  firmly  fastened  to  the  base  ABCD,  upon  which  the 
dial  itself  is  drawn.  The  edge  ab  casts  the  shadow  by  means 
of  which  the  dial  measures  time. 

It  is  necessary  that  the  angle  bac  at  the  base  of  the  gnomon 
be  equal  to  the  terrestrial  latitude  of  the  place  in  which  the 
dial  is  to  be  used.  And  the  gnomon  may  be  designed  easily 
so  as  to  have  the  correct  angle  by  the  method  shown  in 
Fig.  24. 

78 


THE   SUNDIAL 


FIG.  23.    Horizontal  Sundial. 

Draw  the  line  ac  of  any  desired  length,  according  to  the 
size  of  dial  it  is  intended  to  con- 
struct. At  the  point  c  draw  the 
line  cb  perpendicular  to  ac.  The 
proper  length  of  cb  may  be  found 
by  multiplying  the  length  adopted 
for  ac  by  the  factor1  given  in  the  FlG- 24-  Drawing  the  Gnomon. 
following  little  table  for  various  terrestrial  latitudes  : 

TABLE  FOR  CONSTRUCTING  THE  GNOMON 


LAT. 

25° 
30° 
35° 
40° 
45° 
50° 
55° 


FACTOR 

0.466 
0.577 
0.700 
0.839 
1.000 
1.192 
1.428 


Thus,  in  latitude  40°,  if  ac  has  been  made  10  inches  long,  cb  would  be 
8.39  inches. 


1  Note  9,  Appendix. 
79 


ASTRONOMY 

This  having  been  done,  the  gnomon  will  have  the  proper 
angle  at  its  base.  The  construction  of  the  dial  itself  is 
shown  in  Fig.  25.  The  double  line  ac  corresponds  to  the 
line  ac  in  Fig.  24,  the  two  lines  composing  the  double  line  ac 
in  Fig.  25  being  separated  by  the  exact  thickness  of  the 
brass  plate  used  in  making  the  gnomon.  The  gnomon 
must  afterwards  be  fastened  to  the  dial  in  such  a  way  that 
ac  of  Fig.  24  will  come  exactly  upon  ac  of  Fig.  25. 

The  hour  lines  of  Fig.  25  are  drawn  as  follows :  continue 
the  double  line  ac  to  a  point  M,  and  make  the  distance  cM  of 
such  a  length  that  it  will  be  equal  to  the  length  of  ac  multi- 
plied by  the  factor1  given  in  the  following  little  table  for 
various  terrestrial  latitudes : 

TABLE  FOB  CONSTRUCTING  DIAL  LINES 

LAT.  FACTOR 

25°  0.423 

30°  0.500 

35°  0.574 

40°  0.643 

45°  0.707 

50°  0.766 

55°  0.819 

Now  draw  the  long  line  PcQ  of  indefinite  length,  perpen- 
dicular to  ac ;  and  draw  the  two  lines  MN  parallel  to  PQ. 
Draw  the  two  circular  arcs  cc'  with  centers  at  M ,  and  divide 
each  arc  into  six  equal  parts,  giving  the  points  1,  2,  3,  4,  5,  7, 
8,  9,  10,  11.  Draw  lines  as  shown  :  M  1,  M  2,  M  3,  M  10, 
M  11,  etc.,  and  continue  them  to  the  line  PQ,  giving  the 
points  I,  II,  III,  IV,  5',  XI,  X,  IX,  VIII,  7'.  Then  the  lines 
a  I,  a  II,  a  III,  a  IV,  a  V,  a  XI,  a  X,  a  IX,  a  VIII,  a  VII,  as 
shown,  will  be  the  hour-lines  of  the  dial  for  the  several 
hours  of  the  day.  The  six  o'clock  line  is  drawn  from  a  to 
VI,  parallel  to  PQ. 

^Note  10,  Appendix. 
80 


THE   SUNDIAL 


D- 
81 


ASTRONOMY 

The  hour-lines  having  been  drawn  in  this  way,  and  the 
gnomon  fastened  to  the  base  as  already  indicated,  the  whole 
instrument  is  ready  for  use.  When  setting  it  up  in  the 
sunshine,  however,  it  must  be  properly  " oriented,"  or 
turned  around  to  the  correct  position.  This  will  be  the  case 
if  the  line  ac  is  made  to  point  in  the  exact  north-and-south 
direction,  the  end  c  being  toward  the  north.  And  the 
easiest  way  to  orient  the  dial  is  to  turn  it  until  the  shadow 
of  the  gnomon  indicates  the  time  in  accord  with  a  good  watch 
previously  set  to  correct  time. 

But  it  must  not  be  expected  that  the  sundial  will  keep 
pace  accurately  with  the  watch.  For  the  dial  shows  the 
shadow  cast  by  the  actual  visible  sun.  And  as  the  actual 
visible  sun  gives  us  apparent  solar  time  (p.  67),  the  sundial 
must  also  give  apparent  solar  time. 

The  difference  between  this  kind  of  time  and  mean  solar 
time  (p.  71)  is  shown  in  the  following  table  for  various  dates 
in  the  year;  and  this  difference  should,  of  course,  also  be 
considered  when  orienting  the  dial  by  means  of  a  watch. 

TABLE  OF  DIFFERENCES  BETWEEN  SUNDIAL  TIME  AND  MEAN  SOLAR 

TIME 

Jan.        1  Dial  slow    4 m.  July  1  Dial  slow  3 m. 

Jan.      15  "  slow  10  July  15  "  slow  5 

Feb.       1  "  slow  14  Aug.  1  "  slow  6 

Feb.      15  slow  15  Aug.  15  slow  4 

March    1  "  slow  13  Sept.  1  '  correct 

March  15  "  slow    9  Sept.  15  '  fast    5 

April       1  "  slow    4  Oct.  1  '  fast  10 

April     15  "  correct  Oct.  15  '  fast  14 

May       1  "  fast     3  Nov.  1  '  fast  16 

May     15  "  fast     4        C  Nov.  15  '  fast  15 

June       1  "  fast     3  Dec.  1  "  fast  11 

June     15  correct  Dec.  15  fast   4 

We  have  already  stated  (p.  71)  that  the  detailed  explana- 
tions of  these  varying  differences  between  the  two  kinds  of 

82 


THE  SUNDIAL 

solar  time  will  be  found  in  a  later  chapter ;  for  our  present 
purpose  it  is  sufficient  to  use  the  foregoing  tabulation  with- 
out further  comment. 

But  we  must  not  expect  sundial  time  to  agree  exactly  with 
our  watches,  even  after  we  have  made  allowance  for  the 
above  table  of  time  differences.  For  our  watches  indi- 
cate standard  time  (p.  74),  whereas  the  foregoing  table 
merely  corrects  sundial  time  to  make  it  accord  with  mean 
solar  time.  To  ascertain  the  additional  correction  re- 
quired to  transform  the  mean  solar  time  into  standard  or 
watch  time,  we  must  know  the  longitude  difference  of  the 
place  where  the  dial  is  located  from  the  nearest  standard 
meridian. 

For  instance,  New  York,  in  longitude  74°  west  of  Green- 
wich, is  1°  east  of  the  nearest  standard  meridian,  which 
is  in  75°  west  longitude  from  Greenwich.  Therefore  New 
York  local  mean  solar  time  is  later  (or  fast)  of  the  nearest 
standard  meridian  (p.  74).  The  difference  will  be  4m, 
since  1°  must  correspond  to  4m  if  15°  of  longitude  correspond 
to  lh.  It  follows  that  the  sundial,  even  after  the  correction 
from  our  table  has  been  applied,  will  still  always  be  4m  fast 
of  standard  time  as  used  in  New  York.  This  final  difference 
of  4m  should  again  also  be  considered  in  orienting  a  dial  by 
means  of  a  watch. 

The  foregoing  directions  for  making  a  sundial  have  been 
put  in  such  form  that  any  one  can  use  them,  even  if  entirely 
ignorant  of  astronomic  principles.  But  the  knowledge  we 
have  gained  in  Chapter  IV  should  enable  us  to  understand 
the  sundial  much  more  thoroughly.  In  the  first  place,  we 
recall  (p.  40)  that  the  altitude,  or  angular  elevation,  of  the 
north  celestial  pole  above  the  horizon  is  exactly  equal  to  the 
terrestrial  latitude  of  the  observer.  Now  we  have  made  the 

83 


ASTRONOMY 

surface  of  our  dial  level,  and  constructed  the  base  angle 
of  the  gnomon  such  that  the  time-measuring  edge  ab  is  like- 
wise elevated  by  an  angle  equal  to  the  latitude.  And  in 
orienting  the  dial  we  also  turned  it  around  until  the  gnomon 
pointed  exactly  north. 

In  other  words,  the  dial  and  its  gnomon  are  so  arranged 
that  the  edge  ab  of  the  gnomon  points  exactly  at  the  north 
pole  of  the  celestial  sphere.  The  gnomon's  edge  is  therefore 
parallel  to  the  axis  of  the  celestial  sphere ;  since,  as  usual,  we 
may  neglect  the  tiny  radius  of  the  earth  in  comparison  with 
the  infinite  distance  of  the  sphere.  It  follows  that  the 
diurnal  rotation  of  the  celestial  sphere  will  seem  to  take  place 
around  the  edge  of  the  gnomon. 

So  the  sun  each  day  will  also  seem  to  perform  its  diurnal 
rotation  around  the  edge  of  the  gnomon.  Now  we  have  seen 
(p.  68)  that  the  apparent  solar  time  at  any  instant,  or  the 
hour-angle  of  the  visible  sun  at  that  instant,  is  simply  the 
quantity  of  rotation  made  by  the  celestial  sphere  since  the 
sun  was  on  the  meridian.  The  sundial  merely  measures 
this  quantity  of  rotation ;  and  thus  becomes  a  measurer  of 
apparent  solar  time.  When  the  visible  sun  is  on  the  meridian, 
the  shadow  of  the  gnomon  falls,  as  it  should,  on  the  north- 
and-south  line  of  the  dial,  marked  XII.  When  the  quan- 
tity of  diurnal  rotation  is  15°,  or  one  hour,  the  shadow  falls 
on  the  line  marked  I ;  etc.1 

The  accompanying  Plate  4  is  a  photograph 2  of  the  largest 
sundial  ever  built.  It  was  erected  about  1730  by  Jai  Singh  II, 
Maharaja  of  Jaipur,  and  restored  in  1902  by  order  of 
the  Maharaja  Sawai  Madho  Singh.  The  huge  gnomon, 

1  Note  10,  Appendix. 

2  From  The  Jaipur  Observatory  and  its  Builder;    by  Lieutenant  A.  ff. 
Garrett,  R.  E.,  and  Pundit  Chandradhar  Guleri.     Allahabad,  1902. 

84 


THE   SUNDIAL 

containing  stone  stairs,  is  90  feet  high,  and  its  base  is  147 
feet  long.  The  shadow  falls  on  a  great  stone  quadrant 
instead  of  a  level  surface ;  and  the  radius  of  the  quadrant  is 
50  feet.  The  shadow  moves  on  the  quadrant  at  the  rate  of 
two  and  one-half  inches  per  minute. 


85 


CHAPTER  VI 

MOTHER   EARTH 

THERE  was  once  an  old  professor  of  astronomy  who  used  to 
begin  a  lecture  on  "  the  earth  "  by  telling  his  students  that 
the  old  Greek  astronomers  always  assigned  to  the  earth 
the  gender  feminine,  probably  because  she  was  constantly, 
leading  them  astray  in  their  scientific  investigations.  And 
it  must  be  conceded  that  any  one  beginning  to  study  the 
earth  in  its  astronomic  relations  with  the  rest  of  the  universe 
would  find  it  almost  impossible  to  avoid  being  misled  by  his 
early  observations.  In  fact,  the  very  first  thing  we  must 
learn  about  the  earth  is  to  unlearn  almost  everything  we 
ascertain  by  the  actual  use  of  our  eyes. 

For  instance,  if  an  ignorant  person  —  a  person  ignorant  of 
astronomy  —  were  asked  to  examine  the  earth  and  to 
describe  it,  he  would  say  it  is  a  flat  plain,  roughened  with 
hills  and  valleys,  but  still  in  the  main  a  great  plain.  But  an 
astronomer  would  be  compelled  to  ask  him  to  unlearn  this  at 
once,  because  the  earth  is  really  a  big  round  ball  or  globe. 

And  further  direct  examination  of  the  earth  by  this  ignorant 
person  would  lead  him  to  another  fact  which  he  would  con- 
sider certain.  He  would  say  the  earth  is  stiff  and  steady, 
and  that  it  does  not  move.  Another  thing  for  him  to  un- 
learn as  quickly  as  possible ;  for  here  again  is  mother  earth 
a  deceiver,  for  she  is  really  whirling  around  on  an  axis  once 
a  day,  and  also  speeding  along  in  her  annual  orbit  around  the 

86 


MOTHER  EARTH 

sun  at  the  rate  of  about  eighteen  miles  per  second.  And 
she  has  a  number  of  motions  and  wobbles  in  addition  to 
these. 

Now  such  an  imaginary  person  is  by  no  means  to  be 
regarded  as  an  impossibility.  Probably  a  majority  of  those 
who  have  inhabited  the  earth  since  the  beginning  have  been 
thus  ignorant;  possibly  a  majority  of  those  now  living  are 
nearly  as  ignorant.  Some  of  the  greatest  names  of  antiquity 
are  linked  with  conceptions  of  the  universe  quite  at  variance 
with  facts  now  known;  many  of  the  ancient  philosophers 
were  quite  without  knowledge  of  the  earth's  true  motions. 
Pythagoras,  who  lived  in  the  sixth  century  before  Christ, 
or  some  of  his  disciples,  were  perhaps  the  first  to  introduce 
the  idea  of  terrestrial  motion  into  science.  Copernicus,  in 
his  great  work  De  Revolutionibus  (published  1543),  quotes 
the  Pythagorean  philosophers  in  support  of  his  new  theories. 

But  it  is  not  our  purpose  to  trace  the  development  of 
modern  accepted  ideas  as  to  the  earth's  motions  through 
the  vast  literature  of  the  last  four  or  five  centuries ;  we 
shall  confine  our  attention  to  an 
explanation  of  things  as  they  are. 
In  the  first  place,  let  us  consider  the 
rotundity  of  the  earth.  There  are 
a  number  of  convincing  arguments 
to  prove  that  the  earth  is  curved, 
and  not  a  flat  plain  such  as  it  ap- 
pears to  be.  It  has  been  circum-  FlG  26  Cur^uTre  of  the  Earth, 
navigated  many  times,  for  one  thing.  (From  sacroboscus*  sphaera,  Edition 
And  an  even  stronger  proof  of  the 

earth's  curvature  is  furnished  by  the  appearance  of  ships  at 
sea.  When  we  examine  a  vessel  approaching  us  from  a  dis- 
tance (Fig.  26),  we  always  see  the  masts  and  sails  before  the 

87 


ASTRONOMY 

hull  becomes  visible ;  and  this  quite  irrespective  of  the  direc- 
tion from  which  the  ship  is  coming  toward  us.  This  proves 
that  the  earth's  surface  is  curved  —  is  convex  —  in  all 
directions.  It  proves  that  the  surface  of  the  earth  slopes 
downward,  as  it  were,  in  every  direction  from  the  point 
where  the  observer  stands.  And  once  granting  that  the 
earth  is  convex,  its  approximate  sphericity  is  proven  beyond 
a  doubt  by  the  shape  of  the  shadow  it  casts  into  space  on 
the  occasions  when  eclipses  of  the  moon  occur.  A  vast 
number  of  such  eclipses  have  been  observed;  and  always, 
without  exception,  the  edge  of  the  obscured  part  of  the  lunar 
surface  is  curved,  and  curved  as  only  the  shadow  cast  by  a 
spherical  earth  could  possibly  be  curved. 

Next,  as  to  the  earth's  axial  rotation :  how  do  we  know 
that  it  turns  daily  on  an  axis  passing  through  the  terrestrial 
poles  ?  Strong  doubts  existed  on  this  point  at  least  as  late 
as  the  time  of  Galileo,  early  in  the  seventeenth  century. 
Thus  we  may  quote  the  following  from  p.  244  of  Salusbury's 
quaint  translation  of  Galileo's  Dialogue  on  the  Two  Chief 
Systems  of  the  World  (published  by  Galileo  in  1632 ;  Salus- 
bury's translation  published  in  1661)  : 

"Salviati:  'As  in  the  next  place,  to  the  instance  against 
the  perpetual  motion  of  the  earth,  taken  from  the  impossi- 
bility of  its  moving  long  without  wearinesse,  in  regard  that 
living  creatures  themselves,  which  yet  move  naturally,  and 
from  an  inborn  principle,  do  grow  weary,  and  have  need  of 
rest  to  relax  and  refresh  their  members  - 

"Sagredus  (interrupts):  'Methinks  I  hear  Kepler  answer 
him  to  that,  that  there  are  some  kind  of  animals  which  refresh 
themselves  after  wearinesse,  by  rolling  on  the  earth ;  and  that 
therefore  there  is  no  need  to  fear  that  the  terrestrial  Globe 
should  tire,  nay  it  may  be  reasonably  affirmed,  that  it 

88 


MOTHER  EARTH 

enjoyeth  a  perpetual  and  most  tranquil  repose,  keeping  itself 
in  an  eternal  rowling." 

To-day,  as  in  the  time  of  Copernicus  or  Galileo,  the  obvious 
astronomical  arguments  are  not  logically  conclusive.  There 
is  nothing  to  determine  whether  the  diurnal  rotation  of  the 
heavens,  sun,  moon,  and  stars,  is  produced  by  the  sky  turning 
around  the  earth,  or  the  earth  itself  turning  in  the  opposite 
direction  inside  the  sky. 

Fortunately  we  have  now  good  experimental  proof  that 
the  earth  really  turns  on  its  axis  once  in  twenty-four  sidereal 
hours.  But,  strange  to  say,  this  experimental  proof  did  not 
exist  until  1851.  In  that  year  the  physicist  Foucault  per- 
formed a  most  striking  experiment  in  the  Pantheon  at  Paris, 
whereby  it  became  possible  for  the  spectators  to  see  the 
earth,  as  it  were,  actually  turning  under  their  feet.  This 
Foucault  experiment,  as  it  has  since  been  called,  is  not  diffi- 
cult to  perform ;  it  has  been  repeated  by  many  astronomers 
and  physicists  since  the  original  observation  was  made,  and 
always  with  the  same  result,  favorable  to  the  hypothesis  of 
terrestrial  axial  rotation. 

Foucault  suspended  a  very  long  pendulum  consisting  of  a 
heavy  ball  attached  to  a  wire  free  to  swing  in  any  direction. 
The  only  object  in  using  a  pendulum  of  great  weight  and 
length  is  to  diminish  the  disturbing  effects  of  possible 
air-currents  in  the  room,  and  of  other  undesirable  causes 
which  might  make  the  oscillations  of  a  smaller  pendulum 
vary  from  their  theoretically  correct  position. 

When  such  a  perfectly  free  pendulum  is  set  swinging  very 
carefully,  it  will  continue  to  vibrate  back  and  forth,  until  it  is 
finally  brought  to  rest  by  the  friction  of  the  surrounding  air, 
and  the  resistance  to  bending  of  the  wire  by  which  it  is 
suspended.  But  the  direction  in  space  of  the  plane  of  vibra- 

89 


ASTRONOMY 

tion  (the  direction  in  which  the  wire  moves  back  and  forth) 
will  tend  to  remain  constantly  the  same,  because  no  forces 
are  applied  to  the  pendulum  at  right  angles  to  the  direction 
of  its  swing;  and  it  would  require  the  application  of  such 
forces  to  alter  the  direction  in  space  of  the  plane  of  oscillation. 
This  principle,  that  a  free-swinging  pendulum  will  tend  to 
oscillate  in  an  unvarying  direction,  is  the  fundamental 
principle  of  the  Foucault  experiment. 

Now  let  us  suppose  for  a  moment  that  the  experiment  could 
be  performed  at  the  north  pole  of  the  earth.  Suppose  we 
could  there  set  the  pendulum  swinging  in  the  direction  of  the 
star  Arcturus,  for  instance,  and  that  we  marked  on  the 
floor,  under  the  swinging  ball,  the  direction  in  which  the 
oscillations  commenced.  Then,  if  there  were  no  axial  ro- 
tation of  the  earth,  the  pendulum  would  continue  to  swing 
back  and  forth,  exactly  over  the  same  mark  until  it  stopped. 
And  it  would  always  swing  in  the  direction  of  the  star 
Arcturus. 

But  if  the  earth  is  turning  under  the  pendulum,  it  will  carry 
the  mark  on  the  floor  around  with  it.  And  the  pendulum 
still  constantly  continuing  to  swing  toward  Arcturus,  there 
must  result  a  visible  rotation  of  the  mark  on  the  floor  with 
respect  to  the  direction  of  the  pendulum's  swing.  This 
motion  of  the  mark  will  keep  pace  exactly  with  the  terrestrial 
axial  rotation;  and  after  the  earth  has  made  a  complete 
rotation  in  twenty-four  sidereal  hours,  the  mark  must  once 
more  come  exactly  in  line  with  the  direction  of  the  pendu- 
lum's oscillation. 

In  any  latitude  other  than  that  of  the  north  pole,  the  state 
of  affairs  is  not  quite  so  simple.  But  it  is  certain  that  in 
any  latitude  whatever,  if  the  earth  is  perfectly  immobile,  and 
has  no  rotation  of  any  kind,  there  can  result  no  motion 

90 


MOTHER  EARTH 

whatever  of  the  mark  on  the  floor  with  respect  to  the  pendu- 
lum. Once  started  over  the  mark,  the  pendulum  must 
continue  to  oscillate  over  it.  Yet  whenever  and  wherever 
this  experiment  has  been  tried,  large  motions  of  the  mark 
have  been  observed.  Moreover,  and  most  important  of  all, 
the  rate  at  which  the  marks  have  been  observed  to  move  has 
always  been  found  to  agree  accurately  with  the  rate  calcu- 
lated by  theory1  on  the  supposition  that  the  earth  rotates 
on  its  axis  once  in  twenty-four  hours.  The  conclusion  is 
irresistible  that  our  earth  is  really  subject  to  such  a  rotation. 

We  are  not  limited  to  the  Foucault  pendulum  for  an  ex- 
perimental demonstration  of  terrestrial  axial  rotation.  It 
was  pointed  out  by  Newton  that  we  can  test  this  question 
by  the  simple  experiment  of  dropping  a  heavy  object  from 
the  top  of  a  tall  tower,  and  noting  exactly  where  it  falls  upon 
the  earth  beneath.  Newton  had  received  a  letter  (Decem- 
ber, 1679)  from  Hooke,  asking  for  some  "  philosophical  com- 
munication." In  his  reply  he  suggests  the  above  experiment 
and  says  the  falling  body  "will  not  descend  in  the  perpen- 
dicular, but,  outrunning  the  parts  of  the  earth,  will  shoot 
forward  to  the  east  side  of  the  perpendicular." 

It  is  obvious  that  if  terrestrial  rotation  really  exists,  the 
top  of  the  tower  will  move  faster  than  the  bottom  because 
it  is  farther  from  the  center  of  the  earth,  and  so  moves  on  a 
longer  radius.  Therefore,  a  body  dropped  from  the  top 
retains  an  extra  eastward  impetus  in  descending,  and  must 
strike  the  earth  a  little  to  the  east  of  the  spot  directly 
under  the  point  from  which  it  was  allowed  to  fall.  It  would 
not  fall  parallel  to  the  string  of  a  plumb-bob. 

Modern  experiments  on  this  principle,  performed  in  1831, 
were  on  the  whole  inconclusive  in  their  results  because  it 

1  Note  11,  Appendix. 
01 


ASTRONOMY 

was  found  impossible  to  avoid  the  interfering  effects  of  air 
currents,  and  because  the  metal  balls  that  were  allowed  to 
fall  could  not  be  prevented  from  being  deflected  a  little  one 
way  or  the  other  as  a  consequence  of  friction  with  the  air. 
The  errors  introduced  by  these  disturbing  causes  were  large 
enough  to  mask  almost  completely  the  eastward  deflection 
predicted  by  Newton;  but  this  deflection  undoubtedly 
exists  to  the  extent  required  by  theoretic  calculations  based 
on  the  accepted  hypothesis  of  terrestrial  axial  rotation. 

Having  thus  described  the  evidence  which  leads  us  to 
believe  in  the  sphericity  and  diurnal  rotation  of  the  earth,  let 
us  next  consider  the  methods  by  which  its  size  and  weight 
have  been  determined.  Up  to  the  present  we  have  assumed 
as  a  first  approximation  that  the  earth  is  exactly  spherical 
in  form.  Though  this  assumption  is  not  quite  accurate,  we 
shall  continue  it  a  moment  longer,  and  use  it  to  explain  a 
simple  method  of  measuring  the  earth's  size  approximately. 

We  have  but  to  return  to  the  process  of  Eratosthenes 
of  Alexandria  (250  B.C.),  one  of  the  ancients  who  believed 
the  earth  to  be  round.  Eratosthenes  used  a  method  practi- 
cally equivalent  to  setting  up  a  vertical  post,  and  observing 
each  day  the  length  of  its  shadow  cast  upon  a  level  surface. 
He  was  especially  careful  to  measure  the  shadow  when  it  was 
shortest  each  day.  This  occurs,  of  course,  at  noon,  when 
the  sun  is  on  the  meridian.  Furthermore,  the  length  of  the 
short  noon-shadow  is  not  the  same  every  day,  for  a  very 
simple  reason.  We  recall  that  the  sun  always  appears  at 
some  point  in  the  ecliptic  circle  (p.  27),  and  that  during 
about  half  the  year  that  point  is  located  between  the  celestial 
equator  and  the  north  celestial  pole  (p.  43).  During  that 
half-year  there  must  come  a  day  when  the  sun  appears 
in  that  point  of  the  ecliptic  which  is  farthest  north  from  the 

92 


MOTHER  EARTH 


celestial  equator.  This  point  is  called  the  Summer  Solstice ; 
the  sun  reaches  it  on  or  about  June  21  of  each  year;  on 
that  date  we  have  the  longest  day  of  summer ;  the  sun  rises 
higher  in  the  sky  at  noon  than  it  does  on  any  other  date, 
and  the  noon-shadow  of  a  post  is  the  shortest  of  all  the  noon- 
shadows  during  the  year. 

While  the  noon-shadow  will  thus  be  the  shortest  possible 
on  June  21  everywhere  in  the  northern  hemisphere,  it  will  not 
be  equally  short  in  all  places. 
For,  as  shown  in  Fig.  27,  the 
length  of  the  shadow  will  depend 
on  the  angular  distance  of  the 
sun  from  the  zenith  at  noon.  In 
the  figure,  Z  is  the  zenith,  S  the 
sun  on  the  meridian  at  noon, 
BC  the  post,  and  AB  the  length 
of  the  shadow.  In  a  place  where 
the  sun  is  exactly  overhead,  in 
the  zenith,  the  post  will  cast  no 
shadow ;  but  with  the  sun  at  S, 
the  shadow  has  the  length  AB. 
And  the  angular  distance  of  the  sun  from  the  zenith,  or  the 
angle  ZCS,  can  be  found  easily  by  measuring  the  noon- 
shadow  length  AB  together  with  the  height  of  the  post  BC, 
and  then  constructing  a  diagram  like  Fig.  27. 

Now  Eratosthenes  not  only  made  observations  of  this 
kind  at  Alexandria,  but  he  caused  similar  observations 
to  be  made  simultaneously  at  another  place  called  Syene. 
He  was  able  to  assure  himself  that  the  corresponding  obser- 
vations were  really  made  on  the  same  day  by  using  in  both 
places  the  date  when  the  short  noon-shadow  was  the  shortest 
of  the  whole  year. 

93 


Hor/zonfcr/ 


P/ane 


A  B 


FIG.  27.    Length  of  Short  Noon- 
shadow. 


ASTRONOMY 

The  line  joining  Syene  and  Alexandria  was  a  north-and- 
south  line,  or  terrestrial  meridian  of  longitude,  as  we  would 
call  it  to-day.  Eratosthenes  measured  on  the  surface  of  the 
earth,  with  measuring  rods,  the  linear  distance  between  the 
two  places,  and  found  it  to  be,  in  Greek  measure,  5000 
stadia.  By  combining  this  linear  measurement  with  his 
shadow  observations,  he  was  able  to  ascertain  the  size  of 
the  earth,  supposed  to  be  spherical.  Figure  28  shows  how  this 
was  done.  The  circle  represents  the  earth, 
with  Alexandria  and  Syene  situated  at  A 
and  S.  The  zeniths  of  Alexandria  and  Syene 
lie  in  the  directions  of  7,'  and  Z,  respectively. 
The  shadow  observations  showed  that  the 
sun,  on  the  day  when  its  shadow  was  shortest 
at  noon,  was  exactly  in  the  zenith  at  Syene, 
while  on  the  same  day  at  Alexandria  the 
angular  distance  of  the  sun  from  the  zenith 
was  one-fiftieth  part  of  a  circumference,  or 
7°  12,'  as  we  should  call  it  in  modern  angular 

FIG.   28.      Eratos- 
thenes' Measure-  measure. 

Earth  °f  the  Now  the  sun's  distance  from  the  earth  is 
so  great  that  its  rays  falling  on  Alexandria 
and  Syene  may  be  regarded  as  parallel.  Therefore  these 
rays  would  come  down  to  the  point  A  in  a  direction 
parallel  to  ZS ;  and  so  the  angular  distance  7°  12',  measured 
at  A,  is  equal  to  ACS,  the  angle  at  the  earth's  center 
between  terrestrial  radii  drawn  to  Syene  and  Alexandria. 
In  other  words,  Eratosthenes  found  that  5000  linear  stadia, 
measured  on  the  surface  of  the  earth,  correspond  to  one- 
fiftieth  of  the  entire  circumference.  Consequently,  the 
linear  length  of  the  earth's  whole  circumference  must  be 
50  X  5000  stadia,  or  250,000  stadia.  And  from  this 

94 


MOTHER  EARTH 


\Moreuil 


Sourdon 


iidier 


.Arbrede 
1  Boulogne 


|  lonquiere 


Moreuil 


measurement  of  the  circumference 
Eratosthenes  could  find  the  length 
of  the  earth's  radius,  also  in  stadia. 
Unfortunately,  we  do  not  know  the 
length  of  his  stadium  in  modern 
measures,  and  are  therefore  unable 
to  judge  the  precision  of  his  result. 

But  this  old  method  of  Eratos- 
thenes is  to-day  still  in  principle 
the  method  used  for  measuring  the 
earth ;  though  modified,  of  course, 
by  modern  instruments  of  precision, 
and  modern  methods  of  observing. 
Accurately  stated,  the  process  of 
measuring  the  earth,  which  is  called 
Geodesy,  consists  of  two  separate 
and  distinct  operations.  The  first 
corresponds  to  the  measurements 
Eratosthenes  made  on  the  surface 
of  the  earth  between  Syene  and 
Alexandria  for  finding  the  linear 
distance  between  these  two  places. 

Two  suitable  fundamental  points 
on  the  earth's  surface  are  selected, 
and  their  relative  positions,  as  well 
as  the  linear  distance  between  them, 
are  measured  with  the  utmost  pre- 
cision. This  is  accomplished  by 
means  of  a  survey  called  a  geodetic 
triangulation.  First,  a  chain  of 
triangles  is  laid  down  on  the  earth,  as  shown  in  Fig.  29 ;  and 
then,  with  very  accurate  surveying  instruments,  all  their 

95 


iDaittrrurHn 


'Mont  lay 


Pans  ] 


'Malvoisine 
I 

FIG.  29.    Geodetic  Triangula- 
tion. 

(From  Picard's  Degrt  du  Meridien 
entre  Paris  et  Amiens,  Plate  II,  p.  1 16. 
Paris,  1740.) 


ASTRONOMY 

angles,  and  at  least  one  side  of  one  triangle,  are  measured 
with  the  greatest  care. 

The  triangles  are  usually  laid  down  in  such  a  way  that  the 
two  fundamental  points  originally  chosen  are  situated  near 
the  two  ends  of  the  chain  of  triangles,  and  preferably  near 
the  north  and  south  ends.  Then  a  north-and-south  line  is  in- 
serted in  the  survey ;  and  thus  the  process  of  geodetic  triangu- 
lation  finally  furnishes  us  with  the  precise  linear  distance 
by  which  one  of  the  original  points  is  north  of  the  other ; 
or,  in  other  words,  the  modern  equivalent  of  the  5000  stadia 
of  Eratosthenes.  This  distance  is  now  usually  expressed 
in  meters  or  in  feet. 

In  addition  to  the  triangulation,  the  other  operation, 
which  corresponds  to  Eratosthenes'  post  and  shadow  obser- 
vations, is  completed  with  precise  astronomical  instruments 
such  as  will  be  explained  in  a  later  chapter.  For  our  present 
purpose,  it  is  sufficient  to  remark  that  with  the  astronomical 
instruments  in  question  it  is  possible  to  determine  by  observa- 
tion of  the  stars,  and  with  very  high  precision,  the  exact 
terrestrial  latitudes  of  our  two  fundamental  end  points.  This 
having  been  done,  the  difference  of  the  two  latitudes,  so  de- 
termined, gives  us,  in  degrees,  an  arc  corresponding  to  the 
arc  Eratosthenes  measured  with  his  shadows. 

Recurring  to  Fig.  28,  we  may  now  let  the  points  A  and  S 
represent  the  two  end  points  of  the  triangulation,  supposed 
situated  on  a  north-and-south  line,  or  terrestrial  meridian. 
The  survey  gives  the  linear  distance  AS ;  and  the  astronomi- 
cal observation  of  the  latitude  difference  gives  the  corre- 
sponding angle  ACS  at  the  earth's  center.  It  is  then  easy  to 
form  the  following  proportion  :  Angle  ACS  :  360°  : :  linear  dis- 
tance AS :  linear  length  of  entire  circumference. 

By  the  aid  of  this  proportion  we  can  calculate  the  length 

96 


MOTHER  EARTH 


(or  number  of  feet)  in  the  earth's  circumference,  and  thence 
obtain  the  length  of  the  terrestrial  radius.1  It  is  3959  miles 
long. 

It  is  scarcely  necessary  to  remark  that  operations  of  this 
kind  for  determining  the  size  of  the  earth  have  been  repeated 
frequently  at  many  different  parts  of  the  earth's  surface. 
Indeed,  the  importance  of  the  problem  warrants  the  expen- 
diture of  almost  endless  tune  and  trouble  for  its  solution 
with  the  highest  possible  precision. 

And  a  most  interesting  result  has  been  found  from  this 
frequent   repetition   of   Eratosthenes'   method.     The   radii 
obtained  in  different  parts  of  the  earth 
are  not  in  exact  accord.     The  earth  may 
be  considered  spherical  as  a  first  approxi- 
mation,   but    as    a    first    approximation 
only. 

When  we  measure,  for  instance,  the 
number  of  feet  in  an  arc  corresponding 
to  1°  of  latitude  difference  near  the 
equator  of  the  earth,  and  again  in  a  very  high  latitude  near 
the  north  pole,  we  find  the  two  numbers  different.  The 
polar  degree  is  longer  in  feet  than  is  the  equatorial  degree. 
This  can  be  explained  in  one  way  only.  The  earth  is  not 
an  exact  sphere,  but  is  flattened  somewhat  at  the  poles,  so 
that  the  meridian  section  is  shaped  somewhat  as  shown  in 
Fig.  30  (greatly  exaggerated). 

It  is  obvious,  of  course,  that  the  more  flattened  a  circular 
arc  is,  the  longer  must  be  the  radius  of  the  circle.  A  little 
circle  with  a  radius  of  one  inch  will  exhibit  considerable  cur- 
vature even  in  a  very  short  arc ;  but  a  large  circle,  with  a 

1  The  radius  of  a  circle  can,  of  course,  be  computed  easily  from  the  cir- 
cumference by  well-known  mathematical  methods. 
H  97 


FIG.  30.    Flattening  of 
the  Earth. 


ASTRONOMY 

radius  of  100  yards,  will  show  but  very  little  curvature  in  a 
short  piece  of  it.  So  the  curvature  of  our  earth  at  the  poles 
is  like  that  of  a  large  circle ;  near  the  equator  it  is  like  that 
of  a  smaller  circle. 

Now  this  flattening  of  the  earth  at  the  poles  is  exactly 
what  we  should  expect  if  the  earth's  form  has  been  in- 
fluenced by  its  daily  axial  rotation;  and  it  is  certain  to 
have  been  so  influenced.  The  rotation  must  produce  a 
centrifugal  force  which  would  tend  to  make  the  particles 
of  matter  composing  the  earth  move  from  the  polar  to  the 
equatorial  regions.  The  quantity  of  such  motion,  and  the 
consequent  quantity  of  flattening,  must  depend  on  the 
velocity  of  rotation.  If  the  earth  rotated  several  times  as 
fast  as  it  actually  rotates,  we  should  expect  a  considerably 
larger  difference  between  the  polar  and  equatorial  diameters 
of  the  earth. 

Newton  made  an  attempt  to  calculate  the  flattening  of 
the  earth  by  means  of  his  newly  discovered  law  of  gravita- 
tion. But  his  result  was  not  accurate ;  on  account  of  cer- 
tain inherent  difficulties  of  the  problem,  it  can  be  solved 
best  by  actual  observations  rather  than  theoretical  com- 
putations. In  1672,  the  astronomer  Richer  had  already 
made  a  scientific  expedition  to  Cayenne,  and  there  found 
that  his  astronomical  clock,  which  ran  correctly  at  Paris, 
lost  about  two  minutes  daily.  This  was  mainly  due  to  the 
same  centrifugal  force  by  which  the  flattening  of  the  earth 
is  produced.  Richer's  clock  was  a  pendulum  clock.  At 
Cayenne,  near  the  equator,  the  centrifugal  force  must  be 
near  its  maximum.  For  this  force  being  due  to  the  earth's 
motion  of  rotation,  it  will  be  greatest  in  places  near  the 
equator,  which  are  whirling  around  rapidly  in  a  large  circle. 
Places  near  the  pole  are  near  the  rotation  axis,  and  have 

98 


MOTHER  EARTH 

therefore  comparatively  slow  motion  and  moderate  cen- 
trifugal force. 

So  the  centrifugal  force  at  Cayenne,  being  large,  and 
acting  contrary  to  the  gravitational  force  of  the  earth  as  a 
whole,  diminished  that  pull  upon  Richer's  pendulum,  and 
therefore  made  it  oscillate  slower,  so  that  the  clock  "lost 
time.'7 

In  spite  of  Richer's  observation  and  Newton's  calcula- 
tion, many  scientific  men  doubted  the  polar  flattening  of 
the  earth ;  especially  as  certain  French  geodetic  results  did 
not  accord  with  this  theory.  But  in  1735-1744  Maupertuis 
measured  a  meridianal  arc  in  Lapland,  and  Bonguer  and  La 
Condamine  one  in  Peru;  the  comparison  of  these  arcs  left 
no  doubt  of  Newton  having  been  right. 

In  comparatively  recent  years  our  knowledge  of  the 
earth's  true  shape  has  been  extended  greatly  by  entirely 
new  methods  which  we  have  not  yet  described.  The  modern 
applications  of  Eratosthenes'  plan  have  all  involved  trian- 
gulations  extending  in  a  north-and-south  direction  only. 
But  it  should  be  possible  to  employ  with  equal  advantage 
similar  geodetic  surveys  extending  in  an  east-and-west 
direction.  Only,  in  the  latter  case,  the  purely  astronomic 
observations  would  involve  a  determination  of  the  longi- 
tude difference  between  the  two  end  stations  of  the  survey, 
instead  of  their  latitude  difference. 

But  astronomers  had  no  means  of  measuring  longitudes 
with  a  precision  comparable  to  their  measures  of  latitude 
until  the  introduction  of  the  electric  telegraph.  If  the  two 
end  stations  are  telegraphically  connected,  it  is  easy  to 
send  practically  instantaneous  signals  from  one  station  to 
the  other.  By  means  of  these  signals,  accurate  clocks, 
regulated  by  observations  of  the  stars,  and  mounted  at  the 

99 


ASTRONOMY 

two  stations,  can  be  compared,  and  thus  the  time  difference 
(p.  72)  of  the  two  stations  determined  within  a  small 
fraction  of  a  second.  And  the  time  difference  once  known, 
the  corresponding  longitude  difference  is  at  once  obtained, 
since  15°  of  longitude  correspond  to  each  hour  of  time 
difference.  Furthermore,  since  it  has  become  possible  to 
determine  the  terrestrial  radius  by  east-and-west  triangu- 
lations,  it  follows  that  we  can  now  use  equally  well  trian- 
gulations  extending  in  any  direction  whatever,  provided  we 
measure  both  the  latitudes  and  the  longitudes  at  the  two 
end  stations. 

Still  another  and  quite  different  method  of  verifying  the 
precision  of  results  obtained  from  triangulations  has  been 
introduced  in  recent  years.  We  have  seen  that  the  in- 
creased centrifugal  force  near  the  earth's  equator,  acting 
against  the  earth's  gravitational  attraction,  tends  to  diminish 
the  effect  of  the  latter,  and  that  a  pendulum  will  therefore 
swing  more  slowly  near  the  equator  than  it  will  near  the 
poles.  The  quantity  of  this  retardation  can  be  calculated 
accurately  from  the  known  approximate  dimensions  of  the 
earth,  and  its  known  velocity  of  axial  rotation. 

But  when  such  calculations  are  compared  with  actual 
observations  of  pendulums  carried  to  different  places  on  the 
earth,  it  is  found  that  the  retardation  near  the  equator  is 
larger  than  can  be  explained  as  a  result  of  centrifugal  force. 
The  reason  is  obvious.  On  account  of  the  earth's  flattening 
at  the  pole,  the  pendulum  is  actually  farther  from  the  earth's 
center  when  carried  to  the  equator  than  it  is  in  high  northern 
latitudes,  near  the  pole.  As  gravitational  attraction,  accord- 
ing to  Newton's  theory,  diminishes  with  any  increase  of 
distance  from  the  attracting  body,  it  follows  that  the  earth's 
pull  upon  a  pendulum  will  be  a  minimum  at  the  equator. 

100 


MOTHER  EARTH 

Consequently,  we  need  merely  carry  a  pendulum  of  un- 
varying length  to  high  northern  and  to  equatorial  latitudes ; 
and  compare  with  great  accuracy  its  time  of  vibration. 
The  difference,  after  correction  for  the  effects  of  varying 
centrifugal  force,  will  be  a  measure  of  the  variations  in  the 
earth's  gravitational  attractive  force,  and  will  thus  become 
a  measure  of  existing  variations  in  the  length  of  the  earth's 
radius.  Very  elaborate  " pendulum  surveys"  of  this  kind 
have  been  made  in  recent  years,  and  these  verify  the  results 
of  our  latitude  and  longitude  geodetic  triangulations. 

We  may  therefore  regard  the  earth's  true  shape  as  now 
known  with  considerable  accuracy.  But  as  this  accuracy 
has  increased,  with  the  introduction  of  modern  precision, 
minor  irregularities  in  the  earth's  shape  have  been  brought 
to  light.  The  meridians  of  our  planet  are  in  the  main 
ovals,  such  that,  approximately : 

Equatorial  diameter  minus  polar  diameter  _   1 
Equatorial  diameter  295* 

But  these  meridians  are  not  ellipses  of  exact  form.  In 
recent  years  a  new  mathematical  term  has  been  introduced 
by  geodesists  to  describe  the  true  shape  of  the  earth.  They 
call  the  earth  a  Geoid ;  and  a  geoid  is  defined  as  a  surface 
everywhere  perpendicular  to  the  direction  of  the  plumb-bob 
string,  or  the  pull  of  gravity,  and  therefore  everywhere 
coinciding  with  the  mean  surface  of  the  ocean.  The  geoid 
surface  coincides  theoretically  with  the  earth's  surface; 
for  it  includes  the  effects  of  centrifugal  force,  as  well  as  all 
possible  variations  of  the  direction  in  which  terrestrial 
gravity  acts,  and  of  the  pull  which  it  exerts. 

Having  thus  indicated  the  methods  employed  by  astrono- 
mers to  measure  Mother  Earth,  let  us  next  consider  the 
process  of  weighing  her.  And  when  we  begin  to  speak  of 

101 


ASTRONOMY 

weighing  the  earth,  it  becomes  necessary  to  emphasize  the 
distinction  existing  between  the  so-called  mass  of  a  body 
of  any  kind  and  its  weight.  Bodies  have  weight  on  the  earth 
simply  because  of  the  gravitational  pull  of  the  earth  upon 
them.  And  we  have  already  seen  that  this  gravitational 
pull  is  not  everywhere  the  same,  being  greatest  near  the 
poles,  where  the  flattening  of  the  earth  brings  us  nearest  to 
the  center.  Consequently,  we  need  some  kind  of  a  unit, 
analogous  to  a  unit  of  weight,  but  one  that  is  everywhere 
the  same. 

The  unit  of  mass  is  such  a  unit.  The  weight  of  a  body 
is  variable  in  different  places,  but  its  mass  is  everywhere 
the  same.  If  we  can  determine  its  mass  in  one  place,  we 
know  its  mass  everywhere.  For  instance,  if  we  adopt  as  our 
unit  of  mass  a  certain  standard  pound  that  is  preserved  in 
the  United  States  Government  Bureau  of  Standards  in  Wash- 
ington, and  if  we  wish  to  know  the  mass  of  a  certain  stone, 
we  might  carry  it  to  Washington,  and  there  weigh  it  in 
comparison  with  the  standard  pound. 

If  it  weighed  exactly  as  much  there  as  the  standard  pound, 
we  should  say  it  had  a  mass  of  one  pound.  Now  anywhere 
else  on  the  earth  it  would  still  weigh  very  nearly  one  pound, 
because  the  gravitational  attraction  exerted  by  the  earth 
varies  but  little  in  different  localities  on  its  surface,  the 
earth  being  so  very  nearly  an  exact  sphere.  But  if  that 
stone  could  be  carried  to  the  surface  of  the  sun,  where  the 
solar  gravitational  attraction  is  about  28  times  as  great, 
on  account  of  the  sun's  vast  bulk,  it  would  then  weigh  28 
pounds ;  but  its  mass  would  still  be  only  one  pound,  as 
before.  Its  mass  having  been  found  to  be  the  same  as  that 
of  the  standard  pound  in  Washington,  it  would  be  the  same 
everywhere  in  the  universe. 

102 


MOTHER  EARTH 

So  when  we  speak  of  weighing  the  earth,  we  mean,  in 
precise  language,  determining  its  mass.  So  far  as  terrestrial 
man  is  concerned,  there  is  no  exactness  in  speaking  of  the 
earth's  weight,  since  there  is  no  such  thing  as  weight,  except 
in  the  case  of  bodies  situated  on  the  earth's  surface  and 
attracted  by  the  earth.  This  state  of  affairs  is  by  no  means 
objectionable,  because,  for  all  practical  purposes  in  as- 
tronomy, it  is  really  the  mass  of  the  earth  that  we  need  to 
know. 

We  need  to  know  how  strongly  the  earth  exerts  a  gravita- 
tional pull  upon  the  other  planets  in  the  solar  system.  And 
under  Newton's  law  of  gravitation  this  pull  is  proportional 
to  the  earth's  mass.  No  such  thing  as  weight  enters  into 
Newton's  law  anywhere.  According  to  that  law,  two  bodies 
whose  masses  are  M  and  M',  and  whose  distance  asunder 
is  Z),  —  two  such  bodies  attract  each  other  with  a  force  of 
attraction  which  may  be  indicated  by  the  following  simple 
formula: 

Force  of  attraction  =  —=-• 

Stated  in  words,  this  formula  means  that  between  these 
two  bodies  exists  an  attractive  force  which  is  proportional 
to  the  product  of  their  masses,  and  inversely  proportional 
to  the  square  of  the  distance  between  them. 

Various  experimental  methods  have  been  used  to  measure 
the  earth's  mass,  all  depending  on  the  following  principle : 
we  take  some  small  object  on  the  earth's  surface,  and  com- 
pare the  attractive  force  exerted  by  the  earth  upon  that 
object  with  the  corresponding  attractive  force  exerted  upon 
it  by  some  other  large  terrestrial  body  of  known  mass. 
The  attractive  force  exerted  by  the  earth  can,  of  course,  be 
measured  by  weighing  the  chosen  small  object  with  an  ordi- 

103 


ASTRONOMY 


nary  balance ;  that  exerted  by  the  large  object  of  known 
mass  must  be  ascertained  by  means  of  special  experiments. 
But  when  we  thus  know  the  relative  attractive  forces  exerted 

upon  the  same  small  object 
by  the  earth  and  the  terres- 
trial body  of  known  mass, 
we  know  the  relative  masses 
of  the  earth  and  that  body, 
since  attractive  force  is 
always  proportional  to  the 
mass  of  the  attracting  body. 
Thus  we  arrive  at  a  knowl- 
edge of  the  mass  of  the  earth 
in  terms  of  the  large  body  of 
known  mass. 

We  shall  first  describe 
the  so-called  "  Mountain 
Method/'  used  successfully 
by  Maskelyne  in  1774  in 
Scotland.1  He  selected  for 
his  terrestrial  body  of  known 

mass  a  certain  hill  called  Schehallien,2  and  made  a  very 
careful  survey  of  the  region  surrounding  it.  Figure  31 

1  Maskelyne,  "Account  of  Observations  made  on  the  Mountain  Sche- 
hallien for  finding  its  Attraction,"  Phil.  Trans.  Roy.  Soc.  LXV,  Part  II, 
p.  500.     "Redde,  July  6,  1775." 

Hutton,  "Calculations  from  the  Survey  and  Measures  taken  at  Schehal- 
lien, etc.,"  Phil.  Trans.  Roy.  Soc.  LXVIII,  Part  II,  p.  689. 

Maskelyne  and  Hutton  carried  out  their  calculations  in  such  a  way 
that  the  density  or  specific  gravity  of  the  earth  was  made  the  principal 
object  of  their  researches.  We  have  modified  slightly  their  presentation 
of  the  subject,  so  as  to  make  the  earth's  mass  or  weight  the  object  sought. 
The  two  problems  are  identical,  as  we  shall  see  further  on. 

2  To  find  a  hill  suitable  for  his  purpose,  Maskelyne  sent  into  Scotland 
a  certain  Charles  Mason,  who  selected  Schehallien  after  a  long  search. 

104 


FIG.  31.     Mountain  Method  of 
Maskelyne. 


MOTHER  EARTH 

shows  his  method  of  procedure.  PQ  is  a  portion  of  the 
earth's  surface,  here  supposed  spherical,  and  C  is  the  center 
of  the  earth.  SA  and  NB  are  two  plumb-bobs  hung  on 
opposite  sides  of  the  mountain  at  two  principal  stations  of 
the  survey.  The  station  N  was  chosen  nearly  due  north 
of  the  station  S. 

Owing  to  the  gravitational  attraction  of  the  hill,  both 
plumb-bobs  were  deflected  toward  it.  Instead  of  pointing 
toward  the  center  of  the  earth  at  C,  they  pointed  toward 
C",  a  point  situated  between  the  center  C  and  the  surface  PQ. 

Now  it  was  possible  to  ascertain  by  observation  both 
the  angle  C  and  the  angle  C'.  For  the  angle  C  is  simply 
the  latitude  difference  of  the  two  stations  N  and  S,  since 
they  are  on  the  same  terrestrial  meridian,  or  north-and- 
south  line.  And  this  latitude  difference  would  be  one  of 
the  results  furnished  by  the  survey,  which  must  make 
known  the  number  of  feet  N  was  north  of  S.  Then,  know- 
ing the  terrestrial  radius,  the  number  of  feet  corresponding  to 
one  degree  of  latitude  was  known,  and  so  the  exact  number 
of  seconds  of  arc  in  the  latitude  difference  was  also  known. 

On  the  other  hand,  the  angle  C'  was  ascertained  by  means 
of  astronomical  observations  at  the  two  stations  N  and  S. 
It  was  merely  necessary  to  make  the  observations  usual  in 
astronomic  determinations  of  terrestrial  latitude.  It  is 
sufficient  for  our  present  purpose  to  mention  here  one 
peculiarity  of  observations  of  this  kind.  It  is  always  neces- 
sary, in  adjusting  our  instruments,  to  make  use  of  a  plumb- 
bob,  or  its  equivalent,  a  spirit-level,  to  ascertain  the  direc- 
tion of  the  zenith  (p.  36)  directly  overhead. 

This  was  the  same  Mason  who  was  employed  in  1763  by  Lord  Baltimore 
and  Mr.  Penn  to  survey  the  famous  Mason  and  Dixon  line  to  settle  the 
boundary  between  Maryland  and  Pennsylvania  in  the  American  colonies. 

105 


ASTRONOMY 

Ordinarily,  results  obtained  in  this  way  are  correct : 
but,  in  the  present  case,  they  were  rendered  incorrect  by  the 
presence  of  the  neighboring  hill  Schehallien.  The  astro- 
nomical latitudes  determined  at  N  and  S  were  necessarily 
both  erroneous,  and  the  errors  were  equal  to  the  deflections 
of  the  plumb-bobs  at  the  two  stations.  Then,  when  the 
latitude  difference  was  derived  from  the  astronomical  obser- 
vations, it  came  out  as  the  angle  C",  instead  of  the  correct 
latitude  difference  C.  In  other  words,  the  astronomic 
observations  gave  the  latitude  difference  referred  to  the 
false  zeniths  indicated  by  the  plumb-bobs  deflected  by  the 
mountain,  while  the  survey  gave  the  correct  latitude 
difference  C. 

In  the  actual  experiment,  the  difference  between  C  and  C' 
came  out  VI"  of  arc,  a  quantity  large  enough  to  admit  of 
easy  measurement;  and  thus  the  angular  deflection  of  the 
plumb-bobs  became  known.  It  was  next  necessary  to  ascer- 
tain by  measurement  the  mass  or  weight  of  the  hill  itself. 
This  was  accomplished  by  first  computing  its  volume  ap- 
proximately from  the  data  furnished  by  the  survey.  Then 
borings  were  made  into  the  hill,  and  specimens  brought  to 
the  surface.  These  were  tested  to  ascertain  their  "  specific 
gravity,"  or  weight  per  cubic  foot,  as  compared  with  water. 
With  this  information  at  hand  it  was  easy  to  find  the  mass 
of  the  hill. 

The  distance  of  the  hill  from  the  plumb-bobs  being  also 
known,  it  now  became  possible  to  calculate  how  great  must 
be  the  attractive  force  exerted  by  the  hill  on  the  plumb- 
bobs  to  produce  the  observed  deflection  of  12"  appearing 
in  the  difference  C"  —  C.  The  attractive  force  exerted  by  the 
earth  on  the  plumb-bobs  was  ascertained  by  weighing  them 
in  an  ordinary  balance ;  and  thus  Maskelyne  found  the 

106 


MOTHER  EARTH 

relative  attractive  forces  of  the  earth  and  of  the  hill  upon 
the  same  plumb-bobs.  And  the  ratio  of  these  two  attrac- 
tive forces  then  made  known  the  relative  masses  of  the  earth 
and  of  the  hill.  We  have  just  seen  that  the  mass  of  the  hill 
was  ascertained  from  the  borings,  etc. ;  and  so  the  mass  of 
the  earth  finally  became  known,  too.  This  great  classic 
experiment  gave  the  first  knowledge  as  to  the  mass  of  our 
planet. 

Unfortunately,  the  result  was  not  very  accurate ;  the  diffi- 
culties inherent  in  the  measurement  and  testing  of  the  hill 
precluded  the  possibility  of  high  preci- 
sion. Consequently,  a  few  years  later 
(1798),  Cavendish 1  employed  a  method 
which  can  be  entirely  completed  in  a 
laboratory,  and  which,  with  various 
minor  modifications,  has  since  given  us 
all  the  information  we  possess  as  to  the 
earth's  weight  or  mass.  Indirectly,  yet 


just  as  surely  as  if  the  earth  could  be  FlG'32'  Torsion  Balance" 
placed  in  a  gigantic  scale-pan,  is  it  possible  to  weigh  the 
planet. 

The  principal  part  of  the  Cavendish  apparatus  is  called  a 
Torsion  Balance,  shown  in  Fig.  32.  A  very  light  rod  ab 
carries  u  small  metal  ball  at  each  end.  The  rod  is  suspended 
at  its  middle  point  d  by  means  of  a  very  fine  silk  thread  cd 
from  a  fixed  support  c.  In  recent  instruments  the  silk 
thread  is  replaced  by  a  fiber  of  quartz  made  by  fusing  the 
quartz  and  drawing  it  out  to  a  microscopic  fineness. 

The  balance  can  be  set  in  rotation  about  the  supporting 
fiber  cd,  and  will  then  oscillate  backwards  and  forwards  like 
an  ordinary  pendulum,  until  it  is  gradually  brought  to  rest 

1  Phil.  Trans.  Roy.  Soc.,  1798,  p.  469. 
107 


ASTRONOMY 

by  the  continued  friction  of  the  surrounding  air.  During 
these  oscillations  the  rod,  of  course,  remains  horizontal,  being 
exactly  balanced  at  its  middle  point  d. 

Before  explaining  the  use  of  such  a  balance  in  weighing 
the  earth,  it  is  necessary  to  show  how  the  so-called  "  con- 
stant" of  the  balance  itself  may  be  determined.  This 
constant  may  be  called  the  "torsional  constant"  of  the 
balance ;  it  is  a  measure  of  the  quantity  of  force  which 
must  be  applied  to  the  balance  in  order  to  make  it  turn 
about  the  support  cd.  This  quantity  of  force  will,  of  course, 
depend  on  the  thickness  and  stiffness  of  the  fiber  suspension 
cd.  For  when  the  balance  turns,  the  fiber  is  twisted,  and  , 
therefore  the  torsional  constant  will  be  large  if  the  fiber  is 
of  such  a  kind  as  to  resist  a  twisting  effort  quite  strongly. 

The  letter  T  is  used  to  designate  the  torsional  constant 
of  any  given  balance.  Accurately  stated,  T  is  the  quantity 
of  force  required  to  turn  the  balance  through  unit  angle, 
the  said  force  being  applied  to  the  balance  at  unit  distance 
from  the  center  d.  In  our  modern  system  of  units : 

Unit  of  length  is  the  centimeter, 

Unit  of  angle  is  57.3°, 

Unit  of  weight  is  the  gram, 

Unit  of  time  is  the  mean  solar  second. 

Now  it  is  possible  to  determine  the  constant  T  for  any 
given  torsion  balance  by  observing  its  time  of  vibration ; l 
and  this  having  been  done,  we  may  apply  the  balance  to 
our  problem.  For  this  purpose,  it  must  be  mounted  in 
such  a  way  that  its  oscillations  can  be  observed  while  it  is 
under  the  influence  of  the  gravitational  attraction  exerted 
by  a  couple  of  heavy  lead  balls  brought  very  close  to  the 
little  balls  which  are  on  the  ends  of  the  torsion  balance  rod. 

1  Note  12,  Appendix. 
108 


MOTHER  EARTH 


Figure  33  shows  the  apparatus,  the  reader  being  here  sup- 
posed to  examine  it  from  above,  looking  down  upon  it  along 
the  direction  of  the  supporting  fiber  cd  (Fig.  32). 

In  Fig.  33  the  line  ab  shows  the  position  in  which  the 
small  balls  a  and  b  would  finally  come  to  rest  after  oscillat- 
ing, if  the  balance  were  allowed  to  oscillate  quite  undis- 
turbed by  the  proximity  of  the  big  lead  balls.  But  if  these 
latter  are  placed  in  the  position  A'  and  B',  their  gravita- 
tional force  will  attract  the  little  balls  a  and  6,  so  that  the 
final  position  of  rest  will  be  a'b'  in- 
stead of  ab.  And  if  the  big  lead  balls 
are  placed  at  A"  and  Bn ',  the  final 
position  of  rest  will  be  a"b" . 

In  addition  to  the  torsion  balance 
the  apparatus  for  the  Cavendish  ex- 
periment must  therefore  include  two 
big  lead  balls,  together  with  suitable 
mechanical  arrangements  for  trans- 
ferring them  conveniently  from  the 
position  A'B'  to  the  position  A"B" . 
This  having  been  provided,  it  is  pos- 
sible to  ascertain  by  observation  the  distances  a'a"  and 
&'&" ;  and  this,  together  with  our  knowledge  of  Ty  will  tell 
us  the  quantity  of  gravitational  attractive  force  exerted  by 
the  big  lead  balls  upon  the  little  balls  a  and  b. 

But  the  corresponding  attractive  force  exerted  by  the 
earth  upon  these  little  balls  a  and  b  may  be  ascertained 
easily  by  weighing  them  in  an  ordinary  balance,  since  weight 
is  merely  a  result  of  the  earth's  gravitational  attraction. 
Thus  we  return  to  the  principle  used  by  Maskelyne,  and 
which  we  have  already  explained  to  be  fundamental  in  all 
experiments  of  this  kind.  Having  ascertained  separately 

109 


FIG.  33. 


Cavendish  Experi- 
ment. 


ASTRONOMY 

the  attractive  force  exerted  on  the  little  balls  by  the  big 
ones  and  by  the  earth,  we  have  once  more  the  ratio  between 
the  masses  of  the  big  balls  and  the  earth,  since  these  attrac- 
tions are  proportional  to  the  masses  according  to  Newton's 
law.  And  since  masses  are  in  a  sense  only  another  name 
for  weights,  we  have  the  ratio  of  the  earth's  weight  to  that 
of  the  big  lead  balls.1 

The  best  result  obtained  in  this  way  for  the  mass  of  the 
earth,  from  the  average  of  several  modern  repetitions  of 
Cavendish's  experiment,  is : 

6  X  1027  grams. 

The  size,  shape,  and  mass  of  the  earth  having  been  deter- 
mined, it  is  easy  to  calculate  its  average  density  or  specific 
gravity.  This  is,  of  course,  simply  the  average  weight  of  a 
cubic  centimeter  of  terrestrial  material  as  compared  with  a 
cubic  centimeter  of  water.  We  have  merely  to  calculate 
the  earth's  volume  from  its  radius,  which  is  extremely  simple 
if  we  regard  the  earth  as  a  sphere,  and  not  very  difficult, 
even  if  we  take  account  of  the  polar  flattening. 

Knowing  the  earth's  volume,  we  can  then  compute  the 
weight  of  an  equal  volume  of  water,  and  the  ratio  of  the 
weight  of  this  volume  of  water  to  the  weight  of  the  earth 
will  be  the  earth's  average  density.  Thus  we  see  that  the 
problem  of  weighing  the  earth  is  really  equivalent  to  the 
problem  of  determining  the  earth's  density.  (Cf.  p.  104, 
footnote.) 

In  this  way  the  earth's  density  is  found  to  be  about  5.5, 
which  means  that  a  cubic  foot  of  average  terrestrial  material 
weighs  5.5  times  as  much  as  a  cubic  foot  of  water. 

Having  now  discussed  the  methods  of  ascertaining  the 

1  Note  13,  Appendix. 
110 


MOTHER  EARTH 

mass  of  our  earth,  and  the  average  density  of  the  materials 
composing  it,  we  shall  next  consider  for  a  moment  the 
structure  of  the  earth's  core.  Our  knowledge  is  here  neces- 
sarily based  on  theoretical  considerations  only,  it  being 
obviously  impossible  to  penetrate  the  earth's  interior  for 
the  purpose  of  making  actual  observations.  The  deepest 
existing  mines  and  borings  pierce  but  a  very  short  part  of 
the  outer  terrestrial  crust,  when  we  consider  that  the  radius 
of  the  planet  is  about  4000  miles. 

But  such  as  they  are,  these  mines  and  borings  indicate  a 
decided  increase  of  temperature  as  we  go  deeper  into  the 
earth.  The  fact  that  such  temperature  increases  are  always 
found  shows  that  there  must  be  a  steady  supply  of  heat 
from  the  interior ;  if  there  were  not,  the  outer  shell  contain- 
ing the  borings  would  speedily  acquire  a  uniform  tempera- 
ture. And  we  have  further  conclusive  evidence  of  great 
interior  heat  from  the  volcanoes. 

Many  theorists  have  held  in  the  past  that  there  is  a  central 
molten  nucleus  in  the  earth ;  we  now  believe  that  the 
hot  nucleus  is  solid.  It  is  doubtless  quite  hot  enough  to 
be  fused  under  ordinary  circumstances ;  but  at  the  enormous 
pressure  existing  inside  the  earth,  it  is  probably  impossible 
for  any  substance  to  melt,  even  at  a  very  high  temperature. 
The  strongest  argument  for  believing  in  a  solid  earth,  as 
against  a  molten  earth  having  a  thin  solid  exterior  shell, 
is  derived  from  the  phenomena  of  the  tides.  The  tidal 
rise  and  fall  of  the  oceans  is  caused  by  the  gravitational 
attraction  of  the  moon.  If  there  were  but  a  thin  shell  of 
solid  earth,  it  would  be  forced  to  rise  and  fall  also,  for  it 
could  slide  on  the  interior  fluid  mass.  And  if  the  shell  rose 
and  fell  with  the  water,  we  would  not  have  observable  tides 
along  the  coast  lines  of  the  continents.  The  earth's  in- 
ill 


ASTRONOMY 

terior  is  therefore  probably  solid,   with  a  rigidity  about 
equal  to  that  of  steel. 

Under  this  theory  we  must  regard  the  fluid  lava  ejected 
by  volcanoes  as  derived  perhaps  from  minor  " pockets,"  in 
some  way  protected  from  the  usual  pressure,  and  therefore 
containing  molten  matter.  Or  we  may  imagine  that  the 
pressure  of  the  crust  may  be  diminished  materially  at  some 
point  for  a  time,  whereby  the  solid  matter  immediately  under 
that  point  might  suddenly  fuse  and  give  rise  to  an  eruption. 

A  very  remarkable  phenomenon  having  a  certain  bearing 
upon  the  above  theories  is  the  Variation  of  Latitude.  This 
was  first  proved  observationally  by  Kiistner  in  1888,  when 
he  found  that  the  latitude  of  the  Berlin  observatory  was 
subject  to  slight  changes.  In  the  following  year  an  expedi- 
tion was  sent  to  Honolulu,  while  careful  observations  were 
continued  simultaneously  in  Germany.  It  was  found  that 
the  latitude  of  Honolulu  increased  when  the  German  lati- 
tudes decreased,  and  vice  versa. 

Since  terrestrial  latitude  is  merely  angular  distance  from 
the  terrestrial  equator,  it  follows  from  the  above  that  the 
earth's  equator  must  be  swinging  in  some  way.  And  as  the 
equator  is  everywhere  90°  distant  from  the  terrestrial  poles 
at  all  times,  it  follows  that  the  earth's  polar  axis  must  also 
be  in  motion. 

Later  elaborate  observational  researches  have  shown  that 
such  is  really  the  case.  The  earth's  pole  is  really  in  motion, 
though  the  motion  is  quite  small.  A  circle  with  a  radius  of 
50  feet  would  include  all  the  pole's  wanderings  so  far  observed. 
Mathematical  investigations  show  that  this  phenomenon 
indicates  a  solid  but  not  quite  absolutely  rigid  earth,  thus 
affording  a  further  verification  of  the  accepted  theory  as  to 
the  solidity  of  our  planet. 

112 


MOTHER  EARTH 

We  now  pass  from  the  interior  of  the  earth  to  the  part 
which  is  above  the  surface,  —  the  atmosphere.  This  is  a 
mixture  of  various  gases,  principally  nitrogen  and  oxygen, 
with  small  amounts  of  carbon  dioxide,  water  vapor,  and 
various  rare  gases  in  most  minute  quantities.  The  entire 
atmosphere  is  part  of  the  earth,  and  moves  with  it  in  its 
diurnal  rotation  and  annual  orbital  revolution. 

Perhaps  the  most  important  function  of  the  atmosphere 
is  the  distribution  of  sunlight  in  all  directions  by  reflection 
from  the  tiny  particles  in  the  atmosphere.  This  explains 
our  being  able  to  see  objects  on  the  earth  by  the  help  of 
sunlight.  We  cannot  see  such  objects  unless  sunlight  falls 
on  them  in  the  right  direction  to  be  reflected  back  from  the 
object  to  the  observer's  eye.  And  as  the  atmospheric  par- 
ticles reflect  sunlight  in  all  directions,  it  follows  that  some 
light  is  sure  to  fall  on  all  surrounding  objects  in  such  a  way 
as  to  be  reflected  to  our  eye  and  make  the  objects  visible. 

This  same  cause  produces  the  apparent  bright  back- 
ground of  the  sky  in  daytime.  Were  it  not  for  the  atmos- 
phere, the  sky  would  be  dark  in  the  daytime,  as  it  is  at 
night;  and  we  should  see  the  stars  at  all  hours.  And  the 
blue  color  of  the  sky,  as  well  as  the  other  colors  seen  at 
sunset,  etc.,  are  doubtless  a  result  of  prismatic  effects  pro- 
duced by  atmospheric  particles. 

Twilight  is  another  important  phenomenon  due  to  the 
atmosphere.  After  the  sun  has  set  below  the  horizon,  it 
continues  to  illuminate  particles  of  the  upper  atmosphere. 
These  particles  once  more  reflect  the  light,  so  that  a  certain 
diminishing  quantity  of  atmospheric  illumination  continues 
until  the  sun  has  sunk  about  18°  below  the  horizon. 

Another  function  of  the  atmosphere  is  to  act  as  a  kind 
of  blanket  to  retain  solar  heat  upon  the  earth.  The  sun 
i  113 


ASTRONOMY 


sends  us  rays  that  are  practically  all  light-rays.  Rays  of 
this  kind  pass  quite  easily  through  the  atmosphere,  and 
heat  the  earth's  surface.  Then,  at  night,  when  the  earth 
begins  to  radiate  heat  into  space,  it  sends  out  a  kind  of  heat- 
rays  that  pass  through  the  atmosphere  with  the  greatest 
difficulty  only.  Consequently,  the  earth  remains  much 
warmer  than  it  would  otherwise  do ;  and  this  action  of  the 
atmosphere  has  much  to  do  with  making  the  earth  habitable. 
The  phenomenon  is  due  to  a  transformation  of  the  char- 
acter of  solar  rays  by  being  first  absorbed  and  then  radiated 
by  the  terrestrial  surface.  The  water  vapor  in  the  atmos- 
phere is  particularly  effec- 
tive in  this  matter. 

Another,  less  important, 
atmospheric  effect  is  known 
as  Refraction.  Light-rays 
coming  from  any  celestial 
body  must  pass  through 
the  air  before  they  reach 

the  observer.  As  shown  in  Fig.  34,  these  rays  are  bent, 
or  refracted,  as  they  pass  from  the  outer,  and  less  dense, 
parts  of  the  atmosphere  to  the  lower  and  denser  strata. 
The  light  of  a  star  in  the  zenith  at  Z  would  come  straight 
down,  without  change.  For  it  is  a  principle  of  refraction 
that  in  passing  from  any  stratum  to  a  denser  one,  light  is 
not  bent  when  it  is  perpendicular  to  the  strata.  But  if  it 
makes  an  angle  with  the  surfaces  of  the  strata,  it  is  bent 
toward  the  perpendicular. 

Thus  light  coming  from  a  star  at  S  would  travel  through 
the  air  in  a  curve,  and  would  finally  reach  the  observer  at  0 
as  if  it  had  come  in  a  straight  line  from  S'.  The  figure  is, 
of  course,  greatly  exaggerated ;  but  the  effect  of  refraction 

114 


MOTHER  EARTH 

is  to  make  all  the  heavenly  bodies  appear  to  us  nearer  the 
zenith  than  they  really  are.  The  effect  is  greatest  when 
we  observe  near  the  horizon.  Thus  the  sun,  when  setting, 
will  still  be  entirely  visible  after  it  has  passed  below  the 
real  horizon.  At  such  a  time,  too,  the  lower  edge  of  the 
sun,  being  nearest  the  horizon,  is  refracted  more  than  the 
upper  edge.  And  so  the  setting  sun  usually  appears  of  an 
oval  shape  instead  of  round,  as  it  should  be. 


115 


CHAPTER  VII 

THE  EARTH  IN  RELATION  TO  THE  SUN 

IN  the  last  chapter  we  have  discussed  the  earth  as  a  sepa- 
rate astronomic  body,  to  be  measured  and  weighed  without 
special  reference  to  any  other  object  in  the  universe.  We 
have  also  (Chapter  II)  considered  the  earth  to  some  extent 
in  its  relation  to  the  celestial  sphere,  and  found  how  various 
important  points  and  circles  on  that  sphere  correspond  to 
the  terrestrial  poles,  equator,  etc.  Finally,  we  have  made 
use  of  the  plane  of  our  earth's  annual  orbit  around  the  sun, 
extending  it  outward  to  the  celestial  sphere,  to  gain  a  defini- 
tion of  the  ecliptic  circle  (p.  27),  and,  for  the  purpose  of  a 
first  approximation,  we  have  taken  the  earth's  orbit  around 
the  sun  to  be  a  circle,  with  the  sun  at  its  center  (p.  25). 

But  the  real  terrestrial  orbit  around  the  sun  is  a  slightly 
flattened  oval  or  ellipse,  with  the  sun  at  a  point  situated 
near  the  center  of  the  oval,  and  called  the  Focus  of  the 
ellipse.  These  facts  were  first  discovered  by  Kepler,  who 
used  a  method  to  be  described  in  a  later  chapter ;  if  it  were 
necessary  to  establish  their  correctness  to-day,  by  means 
of  observations,  it  would  be  possible  to  do  so  in  a  very 
simple  way. 

The  necessary  observations  would  consist  in  ascertaining, 
on  many  different  dates,  the  exact  position  of  the  point  at 
which  the  sun  appears  projected  on  the  celestial  sphere. 
In  other  words,  we  should  measure  frequently,  with  suitable 
instruments  to  be  described  later,  the  sun's  declination  and 

116 


THE  EARTH  IN  RELATION  TO  THE  SUN 


tic   Clrc/e 


right-ascension  (p.  34) ;  these,  as  the  reader  will  remember, 
define  the  sun's  apparent  position  on  the  celestial  sphere, 
precisely  as  latitude  and  longitude  define  the  position  of 
any  city  on  the  earth's  surface. 

Now  if  we  locate  on  a  celestial  globe  (p.  37)  these  succes- 
sive points  occupied  apparently  by  the  sun  on  various 
dates,  we  shall  find  that  they  all  lie  on  a  single  great  circle 
of  the  celestial  sphere,  which,  as  we  have  already  seen  (p.  27), 
is  called  the  ecliptic 
circle.  And  the  fact 
that  the  sun's  ob- 
served positions  on 
the  celestial  sphere 
thus  all  lie  on  a 
single  great  circle, 
constitutes  an  ob- 
servational proof 
that  the  earth's 
orbit  around  the  sun 
is  really  contained 
in  a  single  plane, 
or  flat  surface. 

Let  us  next,  in 
Fig.  35,  resume  Fig. 
3  (p.  28),  drawing  it,  however,  in  a  slightly  modified  way, 
with  the  earth's  orbit  greatly  enlarged.  But  in  spite  of  this 
enlargement,  the  reader  must  remember  that  the  earth,  sun, 
and  entire  terrestrial  orbit  together  represent  a  mere  dot 
in  comparison  with  the  infinitely  distant  ecliptic  circle  on 
the  celestial  sphere. 

Now,  in  this  Fig.  35,  let  the  large  circle  represent  the 
ecliptic  circle  on  the  celestial  sphere,  and  let  S'  represent 

117 


FIG.  35.     Orbit  of  Earth. 


ASTRONOMY 

various  points  at  which  the  sun  appears  projected,  when 
observed  on  different  dates.  The  true  position  of  the  sun 
in  space  is  always  at  S.  Now  draw  straight  lines  from  these 
observed  points  S'  through  S-y  and  continue  them  to  certain 
other  points  E. 

We  know  that  the  sun  is  projected  on  the  ecliptic  circle 
at  the  points  S'  because  the  earth,  in  its  orbital  motion, 
occupies  successively  the  points  E.  If  we  take  S  as  the 
true  position  constantly  occupied  by  the  sun,  it  follows  that 
when  the  apparent  positions  of  the  sun  on  the  ecliptic  circle 
are  at  the  points  S',  the  earth's  positions  E  will  all  be  some- 
where on  the  extended  lines  S'S.  But  as  yet  we  do  not 

know  where  the  points  E  are 
situated  on  those  extended 
lines  S'S.  We  know  they  are 
somewhere  on  those  lines,  but 
to  know  the  true  shape  of  the 
•of £arth  C2  earth's  orbit  we  must  ascertain 

FIG.  36.    Sun's  Angular  Diameter. 

the  relative  distances  of  the 
various  points  E  from  S  by  a  different  kind  of  observation. 

This  can  be  accomplished  by  measuring,  with  a  suitable 
instrument,  the  Angular  Diameter  of  the  sun  on  the  various 
dates  when  the  positions  S'  were  observed.  To  understand 
what  is  meant  by  angular  diameter,  let  us  imagine  two 
straight  lines,  drawn  from  the  earth  to  two  opposite  points 
on  the  sun's  visible  disk.  Then  the  angle  between  those  two 
lines  is  the  sun's  angular  diameter. 

It  is  quite  evident  from  this  definition  that  the  sun's 
angular  diameter  will  be  greater,  in  proportion  as  the  sun 
is  nearer  to  us.  Figure  36  makes  this  quite  clear.  When 
the  earth  is  near  the  sun,  as  shown  at  E2,  the  angular  diameter 
is  greater  than  when  the  earth  is  farther  from  the  sun,  as  at 

118 


THE  EARTH  IN  RELATION  TO  THE  SUN 

EI.  Consequently,  if  we  have  measured  the  sun's  angular 
diameter  corresponding  to  each  terrestrial  position  E  in 
Fig.  35,  we  can  mark  off  the  relative  lengths  of  the  dis- 
tances SE.  Whenever  the  angular  diameter  was  found  to 
be  large,  we  should  make  SE  proportionately  short,  and 
vice  versa.  The  first  of  the  lines  SE  would  be  made  of  any 
convenient  arbitrary  length,  according  to  the  size  chosen 
for  the  whole  diagram. 

When  all  this  has  been  done,  the  points  E  will  represent 
various  positions  of  the  earth  in  its  orbit.  A  smooth  curve 
can  be  drawn  through  them,  and  it  will  be  found  to  be,  not 
a  circle,  but  a  slightly  flattened  oval  or  ellipse.  The  point 
S,  occupied  by  the  sun,  will  not  appear  at  the  center  of  the 
ellipse,  but  at  the  point  already  mentioned  as  being  situated 
a  little  to  one  side  of  the  center,  and  called  the  focus. 

But  it  is  most  important  to  notice  that  all  this  experi- 
mentation so  far  gives  us  only  the  true  shape  of  the  earth's 
annual  orbit  around  the  sun.  It  tells  us  nothing  whatever 
about  the  actual  size  of  the  orbit  in  miles.  This  could  not 
be  otherwise,  in  the  nature  of  things.  For  up  to  the  present 
we  have  measured  angles  only ;  angular  right-ascensions 
and  declinations,  and  angular  diameters.  And  it  is  a 
mathematical  principle  that  angles  alone  can  never  make 
linear  distances  known.1 

One  more  interesting  fact  might  be  verified  experimentally 
by  the  methods  we  have  just  described.  Referring  again 
to  Fig.  35,  if  the  dates  corresponding  to  the  terrestrial 
positions  E  are  taken  into  consideration,  it  will  be  found 
that  the  line  joining  the  earth  and  the  sun  moves  in  a  very 
peculiar  manner.  This  line  is  called  the  Radius  Vector. 
It  is  clear  that  it  not  only  revolves  around  the  sun  as  a  sort 

1  Note  14,  Appendix. 
119 


ASTRONOMY 

of  pivotal  point,  but  it  also  lengthens  and  shortens,  accord- 
ing to  the  variations  in  the  curvature  of  the  terrestrial  orbit. 

It  will  be  found  that  the  radius  vector,  in  the  course  of 
these  motions  and  changes  of  length,  always  sweeps  over 
equal  areas  in  equal  intervals  of  time.  If  we  take  three 
positions  of  the  earth  E,  such  that  the  time-interval  between 
the  first  and  second  is  equal  to  that  between  the  second  and 
third,  then  the  space  or  area  included  inside  the  orbit 
between  the  first  radius  vector  and  the  second  is  equal  to 
the  corresponding  space  between  the  second  and  third. 
Each  of  these  areas  is  a  kind  of  triangle,  of  which  two  sides 
are  radii  vectores,  and  the  third  side  is  a  bit  of  the  curved 
orbit.  These  facts  were  discovered  by  Kepler  in  1609, 
using,  as  we  have  said,  a  method  of  investigation  quite  dif- 
ferent from  that  here  described. 

Having  now  attained  a  notion  as  to  the  shape  of  the 
terrestrial  orbit,  it  is  possible  to  explain  one  of  the  astronomic 
phenomena  most  important  to  man, — the  Seasons.  What 
is  the  cause  of  summer  heat  and  winter  cold  ? 

For  the  moment  we  shall  consider  the  northern  hemisphere 
only.  At  a  first  glance,  one  might  suppose  that  the  curved 
shape  of  the  earth's  orbit  would  cause  the  seasons.  For 
the  sun  not  being  accurately  at  the  center,  it  must  happen 
that  we  are  nearer  the  sun  when  at  some  particular  point 
of  the  orbit  than  we  are  at  any  other  time.  When  at  this 
point  nearest  the  sun,  called  Perihelion,  the  earth,  as  a 
whole,  does  actually  receive  a  maximum  of  heat.  But  this 
is  masked  so  completely  by  another  phenomenon  that  it  is 
largely  without  effect  in  determining  the  seasons.  In  fact, 
the  date  of  perihelion  occurs  about  January  1  each  year, 
so  that  we  are  actually  nearest  the  sun  in  winter. 

The  temperature  at  any  given  place  on  thej?arth  depends, 

120 


THE  EARTH  IN  RELATION  TO  THE  SUN 

not  on  our  slightly  varying  proximity  to  the  sun,  but  on  the 
relative  duration^  oL  day  and  nigftit.  When  we  have  long 
"days"  and  short  " nights"  ;  when  the  sun  is  shining  on  us 
during  more  than  half  of  each  24-hour  day,  —  then  is  the  time 
to  expect  hot  summer  weather. 

We  have  already  learned  in  Chapter  II  that  half  the 
ecliptic  circle  on  the  celestial  sphere  lies  between  the  celestial 
equator  and  the  north  celestial  pole ;  that  the  sun  is  seen 
in  that  northern  half  of  the  ecliptic  circle  during  about  half 
the  year ;  and  that  during  such  half-year  it  is  above  the 
horizon  daily  for  more  than  twelve  hours.  To  be  more 
precise,  we  found  that  at  the  times  of  the  equinoxes,  about 
March  21  and  September  22,  when  the  sun  appears  to 
cross  the  celestial  equator,  the  days  and  nights  are  equal, 
and  each  is  twelve  hours  long.  But  at  the  solstices  (cf. 
p.  93),  about  June  21  and  December  21,  when  the  sun 
attains  its  greatest  angular  distance  (or  declination)  north 
and  south  of  the  celestial  equator,  —  at  these  solstices 
we  have  the  longest  and  shortest  days  in  the  year,  mid- 
summer day  and  midwinter  day. 

But  there  is  still  another  factor  influencing  this  question 
of  the  seasons  materially.  As  we  have  just  seen,  the  earth's 
surface  is  heated  more  or  less  in  proportion  to  the  length 
of  time  the  sun's  rays  fall  upon  it ;  but  it  is  also  heated  in 
proportion  to  the  directness  with  which  it  receives  those  rays. 
In  summer,  the  sun  is  not  only  above  the  horizon  each  day 
longer  than  in  winter,  but  it  is  also  higher  up  in  the  sky  when 
it  is  above  the  horizon.  Its  rays  therefore  fall  upon  the 
earth  more  nearly  vertically ;  the  sun  not  only  acts  during 
a  larger  number  of  hours,  but  it  also  acts  more  efficiently 
while  the  effect  is  being  produced. 

The   next   important    question   in    connection   with    the 

121 


ASTRONOMY 

seasons  is  to  inquire  as  to  the  date  when  we  may  expect 
the  hottest  day  of  summer.  We  might  at  first  think  it 
should  occur  at  the  time  of  the  summer  solstice,  about  June 
21 ;  and  we  do,  in  fact,  on  that  date  receive  our  maximum 
heat  per  hour  and  per  day.  But  for  a  long  time  after  that 
date  the  days  continue  longer  than  the  nights;  in  each 
24-hour  period  the  earth  is  heated  more  in  the  daytime  than 
it  is  cooled  at  night ;  it  receives  more  heat  than  it  radiates 
away  into  space,  and  is  constantly  becoming:  hotter. 

But  as  this  process  of  increased  heating  continues,  the 
earth,  being  hotter,  acquires  an  increased  capacity  to  give 
up  or  radiate  heat  in  the  night,  because  a  hot  body  radiates 
faster  than  a  cool  body.  At  the  same  time,  the  daylight 
receipt  of  heat  by  the  earth  diminishes  constantly  as  we 
leave  the  solstice  date  in  June.  So  the  daily  accretion  of 
heat  is  diminishing,  because  of  the  shortening  of  daylight; 
the  outgo  is  increasing,  because  of  increased  power  of  radia- 
tion ;  and  so  there  must  come  a  time  when  a  balance  occurs, 
after  which  the  earth  begins  to  become  cooler  again.  In 
the  temperate  regions  of  the  northern  hemisphere  this 
happens  about  August  1,  instead  of  September  22,  the  ap- 
proximate date  of  the  autumnal  equinox,  which  would  be 
the  date  of  balance  if  it  were  not  for^  the  hot  earth's  increasing 
Capacity  to  radiate  heat.  After  August  1  the  night  radia- 
tion begins  to  exceed  the  daily  gain  of  heat,  and  the  earth 
commences  to  cool,  in  anticipation  of  winter. 

In  the  southern  hemisphere  all  these  effects  are  reversed. 
There  the  south  celestial  pole  is  elevated  above  the  horizon 
instead  of  the  north  celestial  pole;  the  southern  half  of 
the  ecliptic  circle  corresponds  to  the  long  days  and  short 
nights,  instead  of  the  northern  half ;  and  midsummer  comes 
in  December  instead  of  June. 

122 


THE  EARTH  IN  RELATION  TO  THE  SUN 

And  there  is  also  another  difference  between  the  two 
hemispheres  which  is  most  interesting.  We  have  already 
mentioned  that  the  earth  is  nearest  the  sun  about  January  1, 
and  that  this  causes  a  slight  increase  of  heat,  which  we  have 
so  far  neglected  to  take  into  consideration.  In  the  southern 
hemisphere  this  little  increase  of  heat  occurs  in  summer,  and 
so  tends  to  make  the  southern  summer  somewhat  hotter 
than  the  northern  summer. 

On  the  other  hand,  the  fact  that  the  radius  vector  sweeps 
over  equal  areas  in  equal  time-intervals  indicates  that 
the  earth  must  move  faster  in  its  orbit  when  near  the  sun 
than  when  farthest  from  the  sun.  Another  reference  to 
Fig.  35  (p.  117)  will  make  this  clear :  when  the  earth  is  near 
the  sun,  the  triangles  have  short  sides,  and  therefore  the 
earth  must  move  through  a  large  angle  in  a  given  time-inter- 
val so  that  the  short  sides  of  the  triangle  may  be  compen- 
sated by  an  increase  in  the  curved  base,  and  the  area  thus 
maintained  unchanged.  It  is  a  principle  of  mechanics  that 
the  orbital  speed  of  any  planet  must  be  greatest  when  it  is 
nearest  the  sun. 

The  effect  of  this  in  the  case  of  the  earth  is  to  make  it 
traverse  the  perihelion  half  of  its  orbit  seven  days  quicker 
than  the  other  half.  In  other  words,  when  the  sun  appears 
in  the  autumnal  equinox  point  in  September,  we  have  to 
wait  only  about  179  days  for  it  to  reach  the  vernal  equinox 
point  in  March.  But  the  other  half  of  the  ecliptic  circle, 
traversed  apparently  by  the  sun  from  March  to  September, 
requires  about  186  days.  These  numbers  may  be  verified 
by  counting  the  days  between  these  pairs  of  dates,  taken 
from  an  almanac. 

It  follows  that  summer  in  the  southern  hemisphere  is  about 
seven  days  shorter  than  summer  in  the  northern  hemisphere ; 

123 


ASTRONOMY 

and  this  just  about  balances  the  increased  heat  of  the 
southern  summer,  which  we  have  just  seen  is  due  to  its 
occurring  in  the  part  of  the  year  when  the  earth  is  nearest 
the  sun.  In  the  northern  hemisphere,  on  the  other  hand, 
summer  occurs  when  the  earth  is  farthest  from  the  sun ;  but 
it  occurs  in  the  long  half-year  of  186  days.  So  there  is 
an  equalization  of  the  summers  in  the  two  hemispheres. 
Both  are  about  equally  hot.  The  southern  has  slightly 
warmer  days  because  of  the  sun's  proximity,  but  it  has 
seven  less  summer  days ;  the  northern  has  slightly  cooler 
summer  days,  but  seven  more  of  them. 

The  case  is  different  with  the  winters,  as  shown  in  the 
following  schedule : 

NORTHERN  HEMISPHERE  SOUTHERN  HEMISPHERE 

Summer     186  days  (far  from  sun)         179  days  (near  sun) 
Winter       179  days  (near  sun)  186  days  (far  from  sun) 

From  this  it  appears  that  the  southern  winter  is  seven 
days  longer  than  the  northern,  and  also  that  the  southern 
winter  days  are  of  the  cooler  kind  on  account  of  increased 
distance  from  the  sun.  So  there  is  no  equalization  of  winter 
between  the  two  hemispheres,  as  there  is  in  summer.  The 
southern  hemisphere  has  a  somewhat  colder  winter  than  the 
northern  hemisphere;  and  the  summers  are  approximately 
the  same  in  both  hemispheres. 

This  interesting  fact  may  be  stated  in  a  slightly  different 
way :  the  diff eren  ce  between  the  average  summer  and  winter 
temperatures  must  be  greater  in  the  southern  than  in  the 
northern  hemisphere.  And  this  presents  a  much  more 
important  aspect  of  the  whole  question.  If  one  hemisphere, 
taking  the  year  as  a  whole,  is  somewhat  colder  than  the  other, 
can  there  not  have  been  a  remote  age  in  the  earth's  past 
history  when  this  difference  was  far  greater  than  it  now  is  ?  — 

124 


THE  EARTH  IN  RELATION  TO  THE  SUN 

great  enough,  perhaps,  to  account  for  the  vast  glaciers  of  the 
geologic  ice-age. 

Of  course  there  is  but  one  way  in  which  the  difference 
could  ever  have  been  materially  greater  than  at  present : 
there  must  have  been  a  time  when  the  terrestrial  orbit 
was  flattened  in  a  greater  degree  than  now,  and  when  the 
sun  was  consequently  much  farther  from  the  center  of  the 
orbit.  But  was  there  ever  such  a  time,  and,  if  so,  what  was 
the  cause  ? 

It  is  an  obvious  fact  that  the  motions  of  our  earth  will 
not  only  be  influenced  by  the  gravitational  attraction  exist- 
ing between  the  earth  and  the  sun,  but  also  by  that  produced 
through  the  pull  of  the  other  planets.  This  latter  effect 
is  small  compared  with  the  solar  effect ;  but  it  is  powerful 
enough  to  bring  about  certain  very  slow  and  somewhat 
irregular  changes  in  the  earth's  orbit  around  the  sun. 

But  all  these  changes  have  one  peculiarity :  all  are  of  the 
kind  mathematicians  call  Periodic.  That  is  to  say,  none 
can  continue  to  act  indefinitely  in  a  single  direction.  Every 
part  of  the  orbit  that  changes  will  change  first  one  way  and 
then  the  opposite  way,  so  that  after  the  lapse  of  sufficient 
ages  of  time,  everything  about  the  orbit  must  return  again 
to  its  original  form  and  condition. 

There  is  thus  a  peculiarly  impressive  perfection  about 
the  operation  of  Newton's  law  of  gravitation  in  the  solar 
system.  No  matter  what  changes  are  destined  to  occur, 
these  changes  will  never  disrupt  the  system  mechanically. 
So  far  as  gravitational  forces  alone  are  concerned,  the  solar 
system  may  endure  forever. 

It  may  be  of  interest  to  give  here  the  principal  orbital 
changes  of  the  above  kind  which  have  been  brought  to  light 
by  the  labors  of  various  mathematicians  following  Newton. 

125 


ASTRONOMY 

1.  The  orbital  flattening  undergoes  slight  changes  with  a 
period  of  64,000  years. 

2.  The  angle  between  the  celestial  equator  and  the  ecliptic 
circle  (cf .  Fig.  6,  p.  35)  varies  slightly,  with  a  period  of  about 
34,000  years. 

3.  The  longest  axis,  or  diameter,  of  the  oval  terrestrial 
orbit  is  slowly  twisting  around  the  sky  with  a  period  of 
108,000  years. 

While  we  are  considering  these  peculiar  variations  of 
long  period  produced  by  the  complicated  action  and  inter- 
action of  gravitational  forces,  it  will  be  of  interest  to  describe 
briefly  the  famous  phenomenon  known  as  the  Precession 
of  the  Equinoxes.  To  make  this  matter  clear,  it  will  per- 
haps be  best  to  call  attention  to  the  methods  probably  used 
in  ancient  times  to  ascertain  by  observation  the  length 
of  the  year.  In  the  first  place,  astronomers  tried  observa- 
tions by  means  of  shadows.  For  instance,  setting  up  a 
vertical  pole,  it  is  easy  to  fix  the  date  when  the  shadow  at 
noon  is  shorter  than  it  is  on  any  other  date.  This  must,  of 
course,  occur  on  the  day  of  the  summer  solstice  (p.  93), 
when  the  sun  appears  highest  in  the  sky.  And  the  sun  will 
then  appear  at  that  point  of  the  ecliptic  circle  which  is 
farthest  north  of  the  celestial  equator. 

By  counting  the  number  of  days  until  the  same  event 
occurs  again,  it  is  possible  to  obtain  an  approximate  value 
for  the  length  of  the  year.  For  the  year  is  simply  the 
period  of  time  required  by  the  sun  to  complete  an  entire 
circuit  in  its  apparent  motion  around  the  ecliptic  circle, 
due  to  the  real  circuit  of  the  earth  in  its  annual  orbit  around 
the  sun.  By  counting  in  a  similar  way  the  number  of  days 
between  two  widely  separated  occurrences  of  the  same  obser- 
vation, it  is  easy  to  find  the  length  of  a  considerable  number 

126 


THE  EARTH  IN  RELATION  TO  THE  SUN 

of  years  joined  together.  In  this  way  Hipparchus  compared 
the  date  of  the  summer  solstice  fixed  by  Aristarchus  of 
Samos  in  280  B.C.  with  his  own  observation  in  135  B.C.,  and 
thus  found  the  number  of  days  in  145  years.  Dividing  this 
by  145,  he  computed  a  very  accurate  value  of  the  average 
length  of  the  year.  It  was  very  nearly  365J  days. 

Another  method  of  ascertaining  the  length  of  the  year 
was  used  by  the  Egyptians  long  before  the  time  of  Hippar- 
chus. They  observed  the  phenomena  called  the  Heliacal 
Risings  of  certain  bright  stars  near  the  ecliptic  circle.  A 
star  is  said  to  have  its  heliacal  rising  where  it  rises  above 
the  horizon  as  near  as  may  be  at  the  time  of  sunrise.  This 
can  occur  only  on  the  date  when  the  sun,  in  the  course 
of  its  apparent  motion  around  the  ecliptic  circle,  happens 
to  appear  near  the  star  in  question.  Star  and  sun  will  then 
rise  together.  By  counting  as  before  the  number  of  days 
until  the  same  event  occurs  again,  it  is  possible  to  ascertain 
how  many  days  the  sun  requires  to  complete  an  apparent 
circuit  of  the  ecliptic  circle  from  a  given  star  back  to  the 
same  star  again. 

But  the  length  of  the  year  obtained  in  these  two  ways  is 
not  quite  the  same.  The  shadow  year  is  a  little  shorter 
than  that  deduced  from  the  method  of  heliacal  risings. 

The  sun,  in  its  apparent  motion,  travels  from  a  given  point 
of  the  ecliptic  back  to  that  point  again  somewhat  quicker 
than  it  proceeds  from  a  given  star  back  to  the  same  star 
again. 

It  is  a  very  singular  thing  that  the  sun  should  thus  move 
along  the  ecliptic  faster  than  it  moves  among  the  stars. 
There  is  but  one  way  in  which  this  could  possibly  occur. 
The  entire  ecliptic  circle,  or  at  least  the  equinox  and  solstice 
points,  must  have  some  kind  of  motion  among  the  stars. 

127 


ASTRONOMY 

In  other  words,  while  the  sun  is  apparently  traveling 
along  the  ecliptic  circle,  that  circle  must  itself  be  moving 
jdightly  in  the  opposite  direction,  so  as  to  accelerate  the  sun/s 
apparent  motion.  Or,  to  be  more  exact,  if  the  sun  starts 
from  the  vernalTequinox  in  its  annual  apparent  motion,  and 
moves  exactly  one  degree  along  the  ecliptic  circle,  it  will 
then  be  a  very  little  more  than  one  degree  distant  from  the 
vernal  equinox.  While  the  sun  was  moving  its  one  degree, 
the  equinox  also  moved  a  tiny  distance  in  the  opposite 
direction ;  so  that  the  distance  from  the  vernal  equinox  to 
the  sun  is  finally  the  sum  of  the  two  motions. 

Astronomers  call  the  kind  of  year  whose  length  may  thus 
be  determined  by  shadows  the  Tropical  Year.  It  is  the 
interval  between  two  successive  apparent  returns  of  the  sun 
to  that  point  of  the  ecliptic  circle  which  is  farthest  north  of 
the  celestial  equator.  When  the  sun  reaches  that  point  in 
its  apparent  course,  it  turns,  and  begins  to  move  southward 
again.  The  point  is  a  turning-point ;  and  the  word  " tropic" 
comes  from  a  Greek  word  meaning  "to  turn." 

Hipparchus  was  able  to  measure  also  with  considerable 
precision  the  length  of  the  other  year, — the  period  of  time 
required  by  the  sun  to  move  in  its  apparent  course  along  the 
ecliptic  from  a  given  star  back  to  that  star  again.  This 
kind  of  year,  from  its  relation  to  the  stars,  is  called  the 
Sidereal  Year. 


The  difference  between  the  two  kinds  of  year  is  about 
twenty  minutes,  the  tropical  year  being  the  shorter.  Hip- 
parchus explained  the  difference  correctly  as  a  consequence 
of  the  annual  motion  of  the  vernal  and  autumnal  equinox 
points.  Now  these  points  are  merely  the  intersections  of 
the  ecliptic  circle  and  the  celestial  equator  on  the  celestial 
sphere.  If  they  are  in  motion,  such  motion  may  be  caused 

128 


THE  EARTH  IN  RELATION  TO  THE  SUN 

by  a  change  in  position  of  the  ecliptic  circle,  or  the  celestial 
equator,  or  both.  Hipparchus  was  able  to  show  that  the 
effect  is  produced  bv  a  slight  motion  of  the  celestial  equator, 
the  ecliptic  remaining  practically  unchanged.  The  celestial 
equator  is  moving  in  such  a  way  as  to  cause  the  equinoxes 
(its  points  of  intersection  with  the  ecliptic  circle)  to  move 
along  the  ecliptic  circle  very  slowly. 

Hipparchus  had  no  difficulty  in  satisfying  himself  that 
the  ecliptic  circle  did  not  itself  change,  and  that  only  the 
equator  and  the  equinox  points  were  in  motion.  For 
his  star  observations  showed  that  all  the  fixed  stars  main- 
tained constantly  unchanged  angular  distances  from  the 
ecliptic  circle  on  the  sky;  which  could  not  have  been  the 
case  if  that  circle  was  itself  in  motion.  But  the  stars  did 
change  their  angular  distances  (declinations)  from  tiie 
celestial  equator. 

In  fact,  Hipparchus  discovered  these  phenomena  first 
from  his  star  observations,  which  he  compared  with  those 
of  Timocharis  and  Aristyllus,  made  about  150  years  before 
his  day.  From  this  comparison  he  ascertained  the  quantity 
of  motion  of  the  equinoxes,  and  thence  computed  the  differ- 
ence in  length  between  the  tropical  and  sidereal  years.  The 
length  of  the  tropical  year  he  found,  as  we  have  seen,  by 
means  of  shadow  observations.  The  length  of  the  sidereal 
year  he  then  calculated  by  adding  to  the  length  of  the  tropi- 
cal year  the  difference  between  the  two  as  he  had  com- 
puted it.  This  was,  and  is,  the  best  method  of  procedure, 
as  the  length  of  the  sidereal  year  cannot  be  observed  directly 
with  high  precision.  It  was  Hipparchus  who  named  the 
motion  of  the  equinoxes  Precession. 

It  is  possible  to  explain  the  cause  of  precession  by  the  aid 
of  Newton's  law  of  gravitation.  We  have  already  found 

K  129 


ASTRONOMY 


that  the  earth  is  not  truly  spherical,  but  that  it  is  somewhat 
flattened  at  the  poles.  This  amounts  in  effect  to  a  spherical 
earth,  with  a  girdle  of  protuberant  material  surrounding 
the  equator.  In  other  words,  the  earth  has  its  biggest 
girth  around  the  terrestrial  equator.  Figure  37  is  intended 
to  illustrate  the  existing  state  of  affairs.  It  shows  the 
spherical  earth,  with. its  north  pole  (N.  P.),  its  equatorial  pro- 
tuberance, and  the 
planes  of  the  equator 
and  ecliptic. 

Now  the  sun  and 
moon  both  exert  a 
gravitational  attrac- 
tion upon  the  earth, 
and  also  upon  its 
equatorial  protuber- 
ance. And,  as  we 
have  already  seen, 
the  sun  is  always  in 
the  plane  of  the 

ecliptic ;  we  may  add  as  a  fact  that  the  moon  also  happens 
to  pursue  an  orbit  that  never  goes  very  far  from  the  same 
plane.  But  the  lunar  and  solar  attractions  affect  most 
strongly  that  part  of  the  protuberant  ring  which  is  nearest 
to  them.  This  tends  to  tip  over  the  protuberant  ring  into 
the  plane  of  the  ecliptic.  If  no  other  forces  were  at  work, 
the  earth  (Fig.  37)  would  simply  revolve  around  an  axis 
perpendicular  to  the  paper  and  passing  through  the  earth's 
center  C,  until  the  equatorial  plane  had  been  brought  into 
coincidence  with  the  ecliptic  plane. 

The  force  which  prevents  this  rotation  is  due  to  the 
diurnal  turning  of  the  earth   on  its  axis.     The  earth  is 

130 


/Earth 


FIG.  37.    Precession. 


THE   EARTH   IN  RELATION  TO  THE   SUN 

trying  to  turn  around  two  axes  at  once,  —  its  rotation  axis 
through  the  north  and  south  poles,  and  the  other  axis  we  have 
just  mentioned.  The  result  is  to  produce  what  is  called 
in  the  science  of  mechanics  a  composition  of  rotations. 
This  leaves  the  earth  turning  around  its  regular  rotation  axis 
once  daily,  but  makes  that  axis  itself  move  in  space  in  such 
a  way  that  the  celestial  pole  on  its  extended  end  revolves 
slowly  on  the  sky  in  a  circle  around  a  fixed  center  called  the 
Pole  of  the  Ecliptic.  This  is  a  point  on  the  celestial  sphere 
90°  distant  from  every  point  of  the  ecliptic  circle.  The 
celestial  pole  being  merely  the  prolongation  of  the  earth's 
rotation  axis,  to  the  celestial  sphere,  and  the  rotation  axis 
being  set  in  motion  by  the  composition  of  rotations,  the 
celestial  pole  must  evidently  move  on  the  sky.  The  pole 
of  the  ecliptic  remains  unmoved,  because,  as  Hipparchus 
found,  the  ecliptic  does  not  itself  change.  But  the  celestial 
pole,  and  consequently  the  celestial  equator,  are  both  subject 
to  this  precessional  motion. 

The  angular  radius  of  the  circle  in  which  the  celestial  pole 
revolves  on  the  sky  around  the  ecliptic  pole  is  equal  to  the 
angle  between  the  celestial  equator  and  the  ecliptic  circle, 
which  is  about  23  J°.  A  complete  revolution  of  the  one 
pole  around  the  other  requires  about  25,800  years;  for  the 
annual  precession  of  the  equinoxes  upon  the  ecliptic  circle 
is  50.2"  and  a  complete  revolution  of  the  pole  must,  of 
course,  correspond  to  a  complete  revolution  of  the  equinoxes. 
And  we  have  : 

=  25,800,  approximately. 


It  must  not  be  supposed  that  this  precessional  motion 
proceeds  with  perfect  uniformity,  for  there  are  various  causes 
of  inequality.  When  the  sun  appears  at  the  equinoctial 

131 


ASTRONOMY 

points  in  March  and  September,  it  is  for  the  moment  also 
in  the  celestial  equator,  because  the  two  circles,  ecliptic 
and  equator,  cross  at  the  equinoctial  points.  At  such  times 
the  sun  does  not  tend  to  tip  the  earth's  equator.  But  at 
the  time  of  the  solstices,  when  the  sun  is  far  from  the  equato- 
rial plane,  it  has  its  maximum  tipping  effect.  The  moon's 
effect  is  even  more  complicated,  on  account  of  certain 
periodic  changes  in  the  position  of  the  moon's  orbit.  Thus 
the  actual  precessional  circle  marked  out  on  the  sky  by  the 
celestial  pole  really  resembles  a  sort  of  wavy  line,  having 
about  1400  principal  waves  in  an  entire  circuit  of  25,800 
years.  These  waves  are  called  the  Nutation,  or  nodding,, 
of  the  terrestrial  axis. 

An  interesting  consequence  of  precession  is  its  effect  on 
the  seasons  in  the  northern  and  southern  hemispheres.  We 
have  seen  that  the  southern  hemisphere  is  now  on  the  whole 
colder  than  the  northern.  But  after  half  a  precessional 
cycle  has  elapsed,  the  northern  will  be  the  colder  hemisphere. 
Thus  the  astronomical  explanation  of  the  geologic  ice-age 
is  made  possible.  For  the  ice-cap  was  in  the  northern 
hemisphere :  it  must  have  been  formed  at  a  time  when 
precession  made  the  northern  hemisphere  the  colder  one, 
and  when,  coincidently,  the  summer  and  winter  halves  of 
the  year  were  unequal  by  much  more  than  the  present 
difference  of  seven  days,,  on  account  of  the  periodic  change 
of  the  earth's  orbital  flattening  (p.  126). 

Another  important  result  of  precession  is  the  fact  that 
the  celestial  pole  is  not  always  near  our  present  pole  star. 
This  star  is  now  about  1J°  distant  from  the  true  celestial 
pole ;  in  the  time  of  Hipparchus  it  was  12°  distant ;  12,000 
years  hence  Vega  will  be  our  nearest  pole  star;  and  4000 
years  ago  «  Draconis  was  the  pole  star.  This  is  well  shown 

132 


PLATE  5.     Precessional  Motion  of  the  Pole, 


THE  EARTH  IN  RELATION  TO  THE  SUN 

in  the  accompanying  Plate  5,  reproduced  from  Hevelius' 
Prodromus  Astronomies,  Gedani  (Dantzig),  1690.  It  contains 
the  constellation  Draco,  as  drawn  by  Hevelius,  inclosed  in 
the  precessional  circle,  having  the  pole  of  the  ecliptic  at 
its  center.  The  pole  star  appears  at  the  end  of  the  long 
tail  of  Ursa  Minor,  the  Little  Bear.  The  circle  is  divided 
into  degrees,  and  it  indicates  that  Hevelius  observed  the  pole 
star  at  an  angular  distance  of  4°  from  the  celestial  pole, 
which  is  situated  at  the  lowest  point  of  the  circle.  This  is 
in  very  close  agreement  with  the  theory  of  the  precessional 
motion  of  the  pole,  as  explained  above. 

Of  peculiar  interest,  also,  in  this  connection,  are  the 
theories  held  by  Egyptologists  as  to  the  date  of  construction 
of  the  great  pyramid.  In  that  pyramid  there  is  a  long 
passage  pointing  due  north,  and  elevated  above  the  horizon- 
tal at  exactly  the  right  angle  to  view  «  Draconis  when  it  was 
the  pole  star.  There  can  be  little  doubt  that  this  passage 
was  purposely  so  built ;  and  there  is  therefore  little  doubt 
left  as  to  the  approximate  age  of  the  pyramid. 

There  are  still  several  other  details  in  connection  with 
the  relation  of  earth  and  sun  that  we  must  consider  here. 
For  instance,  we  recall  (p.  71)  that  astronomers  use  a 
mean  solar  day  and  a  mean  sun  corresponding  as  accurately 
as  possible  to  the  actual  performances  of  the  real  visible 
sun.  It  is  now  possible  to  make  this  relation  between  the 
mean  sun  and  the  real  sun  a  little  clearer. 

Since  the  length  of  the  mean  solar  day  represents  the 
average  of  all  the  actual  solar  days,  it  is  evident  that  the 
mean  sun  must  be  sometimes  in  advance  of  the  actual  sun, 
and  sometimes  behind  it.  The  difference  between  the  two 
suns  cannot  continue  to  increase  indefinitely;  as  a  matter 
of  fact,  the  extreme  value  of  the  difference  is  sixteen  minutes. 

133 


ASTRONOMY 

In  other  words,  mean  solar  time  may  be  as  much  as  sixteen 
minutes  fast  or  slow  of  apparent  solar  time.  The  difference 
between  the  two  kinds  of  solar  time  is  called  the  Equation  of 
Time.  The  equation  of  time  at  any  moment  is  defined,  then, 
as  the  quantity  of  time  we  must  add  to  the  apparent  solar  time 
at  that  moment,  to  make  it  equal  to  the  mean  solar  time. 

There  are  two  principal  causes  producing  the  equation  of 
time.  The  first  has  already  been  mentioned  repeatedly. 
The  earth  does  not  move  in  its  orbit  with  uniform  velocity, 
but  travels  most  rapidly  near  perihelion  (pp.  70,  123). 
Consequently  the  sun,  projected  on  the  sky  at  its  various 
apparent  positions  in  the  ecliptic  circle,  also  appears  to 
move  other  than  uniformly  in  that  circle.  This,  of  course, 
puts  the  real  sun  in  advance  of  the  mean  sun  at  times,  and 
behind  it  at  other  times. 

But  even  if  the  real  sun  were  projected  upon  the  ecliptic 
circle  with  uniform  motion,  still  there  would  not  result  an 
equality  of  the  actual  solar  days.  A  reference  to  a  celestial 
globe,  or  to  Fig.  6,  p.  35,  shows  that  there  is  a  variable  angle 
on  the  celestial  sphere  between  the  ecliptic  circle  and  the 
celestial  equator.  At  the  equinox  points,  where  the  two 
circles  cross,  there  is  an  angle  of  23 £°  between  them;  but 
at  the  solstices,  where  the  distance  between  the  two  circles 
is  greatest,  they  are  practically  parallel  for  a  short  distance. 

Therefore  even  uniform  motion  in  the  ecliptic  would  not 
give  uniform  motion  when  projected  on  the  equator.  But  it 
would  require  uniform  apparent  motion  of  the  sun  on  the 
celestial  equator  to  produce  equality  of  the  actual  solar  days. 
For  we  have  seen  (cf.  p.  69)  that  the  sun  would  have  to 
move  exactly  the  same  distance  on  the  equator  each  day  to 
make  all  the  apparent  solar  days  exceed  the  unvarying  side- 
real day  by  exactly  the  same  amount.  To  repeat,  then, 

134 


THE  EARTH  IN  RELATION  TO  THE  SUN 

the  two  causes  of  the  equation  of  time  are :  first,  variable 
motion  of  the  earth  in  its  orbit,  producing  variable  apparent 
motion  of  the  sun  in  the  ecliptic  circle;  second,  variable 
angle  between  the  ecliptic  circle  and  the  celestial  equator. 

We  have  already  given  (p.  82)  a  table  of  the  equation 
of  time  for  various  dates  in  the  year.  It  there  appears  as  a 
table  of  errors  of  the  sundial,  because  the  dial  keeps  apparent 
solar  time  by  the  shadow  of  the  actual  visible  sun,  and  must 
be  corrected  by  the  amount  of  the  equation  of  time  to  make 
it  conform  to  mean  time. 

An  interesting  and  frequently  misunderstood  consequence 
of  the  equation  of  time  is  the  inequality  of  the  mornings  and 
afternoons  at  certain  dates  in  the  year.  Morning  begins 
at  sunrise  and  ends  at  noon.  Afternoon  begins  at  noon 
and  ends  at  sunset.  Now  sunrise  and  sunset  occur 
when  the  actual  visible  sun  appears  or  disappears  at  the 
horizon ;  by  convention,  noon  occurs  when  the  mean  sun  is 
on  the  meridian.  Thus  the  morning  will  be  shortened  and 
the  afternoon  lengthened  by  the  amount  of  the  equation  of 
time,  or  vice  versa.  The  difference  on  any  date  will  be  twice 
the  equation  of  time  on  that  date. 

In  February  the  afternoons  are  about  half-an-hour  longer 
than  the  mornings;  in  November,  they  are  half-an-hour 
shorter;  on  account  of  this  effect  of  the  equation  of  time. 
Furthermore,  we  have  seen  (p.  74),  when  considering  stand- 
ard time,  that  the  times  in  actual  use  in  certain  places 
may  differ  from  their  proper  mean  solar  times  by  as  much  as 
half-an-hour.  This  again  affects  the  difference  between  the 
morning  and  afternoon  by  twice  its  amount,  or  a  full  hour. 
On  the  dates  mentioned  above,  it  may  happen  that  this 
hour  is  added  to  the  half-hour  arising  from  the  equation 
of  time;  so  that  on  certain  dates  and  in  certain  places 

135 


ASTRONOMY 

morning  and  afternoon  may  differ  as  much  as  an  hour  and 
a  half.  It  is  easy  to  find  an  example  in  the  ordinary  almanacs. 
Thus  for  November  20,  1913,  the  almanac  gives  the  standard 
times  of  sunrise  and  sunset  at  Detroit  as  6.28  A.M.  and 
4.7  P.M.  This  makes  the  forenoon  5h  32m  long,  while  the  after- 
noon lasts  only  4h  7m.  The  difference  is  nearly  an  hour 
and  a  half. 

There  now  remains  but  one  more  phenomenon  of  impor- 
tance requiring  attention  in  connection  with  our  earth's 
annual  motion  around  the  sun.  It  is  called  Ijie  Aberration 
of  Light,  and  was  discovered  by  James  Bradley,  astronomer 
royal  of  England,  in  1728.  Bradley  had  been  making  some 
very  precise  observations  of  the  declinations  (p.  34)  of  cer- 
tain stars,  and  had  found  that  observations  made  six  months 
apart  could  not  be  brought  into  agreement.  There  was  a 
slight  displacement  of  the  stars  on  the  sky  at  the  end  of  six 
months ;  after  the  lapse  of  a  whole  year  they  were  back 
again  in  their  old  places. 

This  matter  puzzled  Bradley  greatly;  for  a  long  time 
he  was  quite  unable  to  find  any  satisfactory  explanation. 
Finally  he  came  upon  the  solution  of  the  problem  while 
he  was  sailing  in  a  small  boat  on  the  Thames.  At  the 
mast-head  of  the  boat  was  a  pennant ;  and  Bradley  noticed 
that  whenever  the  boat  changed  its  course  in  tacking,  the 
pennant  changed  its  direction  a  little  with  respect  to  the 
river  bank.  There  seemed  no  reason  for  this,  because  the 
wind  was  quite  steady  in  direction. 

Then  it  occurred  to  him  that  the  boat's  own  motion  in- 
fluenced the  pennant.  Its  position  would  be  determined 
by  a  combination  of  the  wind's  velocity  and  direction,  to- 
gether with  the  boat's  speed  and  direction  of  motion.  And 
he  saw  at  once  that  light  coming  to  us  from  a  star  would 

136 


. 

THE  EARTH  IN  RELATION  TO  THE  SUN 

seem  to  come  in  a  direction  similarly  depending  on  the 
true  direction  of  the  star,  and  the  light's  velocity  combined 
with  the  direction  and  velocity  of  our  terrestrial  orbital 
motion.  The  earth  is  here  the  boat;  and  the  aberration 
of  light  was  explained. 

There  is  another  familiar  explanation  which  may  make 
this  phenomenon  clearer.  Imagine  a  person  standing  per- 
fectly still  in  a  rain  storm  on  a  windless  day.  The  drops 
will  seem  to  fall  perpendicularly  downward;  but  if  the 
person  runs  rapidly,  they  will  strike  him  in  the  face,  precisely 
as  if  they  were  coming  down  in  a  slanting  direction.  The 
drops  will  seem  to  come  towards  the  runner ;  more  exactly 
stated,  the  direction  from  which  the  drops  seem  to  come 
will  be  thrown  forward  in  the  direction  of  the  running  oB- 
server's  motion. 

In  a  similar  way,  the  direction  from  which  starlight  seems 
to  come  is  thrown  toward  that  point  on  the  sky  toward 
which  the  terrestrial  motion  is  for  the  moment  aimed . 
We  see  the  star  a  little  too  near  that  point.  But  the  earth 
moves  in  a  nearly  circular  orbit,  and  so  is  constantly  chang- 
ing the  direction  of  its  motion.  Therefore  the  aberrational 
change  in  the  stars'  positions  is  also  constantly  and  similarly 
changing  its  direction.  The  final  result  is  to  make  each 
star  seem  to  describe  a  little  closed  curve  on  the  sky,  which 
is  a  sort  of  miniature  copy  of  the  terrestrial  orbit  around  the 
sun.  This  little  aberrational  curve  is,  of  course,  different  for 
different  stars,  depending  on  their  positions  in  the  sky 
with  respect  to  the  earth's  orbit.  And  the  reason  why 
these  aberrational  curves  are  so  small  is  that  the  velocity 
of  light  is  very  large  compared  with  the  earth's  linear  velocity 
in  its  orbit.  For  if  light  moved  instantaneously,  or  if  the 
earth  had  no  motion,  there  would  be  no  aberration. 

137 


CHAPTER  VIII 

THE    CALENDAR 

PERHAPS  the  chief  duty  of  astronomers  has  always  been 
the  orderly  measurement  of  time ;  not  merely  short  intervals 
such  as  the  hour  and  minute,  but  also  the  much  longer  periods 
represented  by  months  and  years.  For  the  latter  purpose 
various  calendars  have  been  devised.  The  most  ancient 
were  doubtless  based  on  the  motions  of  the  moon,  and 
were  consequently  very  irregular  and  complicated.  It  will 
not  be  of  interest  to  trace  their  development  beyond  the  year 
45  B.C.,  when  Julius  Caesar  put  in  force  at  Rome  the  form  of 
calendar  which  bears  his  name,  and  which  had  been  arranged 
for  him  by  the  Greek  astronomer  Sosigenes  of  Alexandria. 

The  first  thing  to  understand  about  a  calendar  in  the 
modern  sense  is  that  every  date,  such  as  Wednesday, 
August  27,  1913,  is  composed  of  four  different  constituent 
parts  :  the  day  of  the  week,  the  day  of  the  month,  the  name 
of  the  month,  and  the  number  of  the  year.  We  may  then 
define  the  fundamental  problem  of  the  calendar  thus  : 
having  given  any  three  of  these  constituent  parts  of  a  date, 
to  find  the  fourth.  This  is  the  problem  we  shall  solve  in  the 
present  chapter,  both  for  the  Julian  calendar  of  Caesar, 
and  the  modern  Gregorian  calendar,  now  in  general  use. 
This  calendar  was  named  after  Pope  Gregory  XIII,  by  whose 
orders  it  was  introduced  in  1582,  though  it  did  not  receive 
recognition  in  England  until  1752,  and  is  not  yet  used  in 
Russia. 

138 


THE   CALENDAR 

Our  fundamental  problem  may  present  itself  in  several 
different  forms.  For  instance,  an  important  event  in  Ameri- 
can history  happened  on  March  4,  1865 ;  on  what  day  of  the 
week  did  it  occur  ?  This  event  was  the  second  inauguration 
of  Abraham  Lincoln  as  President. 

This  same  event  suggests  a  good  illustration  of  another 
form  in  which  our  problem  may  present  itself.  Presidents 
of  the  United  States  are  always  inaugurated  on  March  4, 
at  intervals  of  four  years ;  and,  with  rare  exceptions,  in  years 
following  a  " leap-year."  In  what  years  during  the  twentieth 
century  will  these  inauguration  dates  fall  on  Sunday  ? 

A  third  form  of  the  problem  might  be  as  follows :  an  old 
letter,  of  great  historic  interest,  happens  to  have  its  date 
blurred  so  as  to  be  partly  illegible.  Suppose  we  can  read, 
however,  that  it  was  written  in  a  certain  year,  and  on  the 
17th  day  of  the  month.  It  also  appears  from  some  re- 
mark in  the  letter  itself  that  it  was  written  on  a  Thursday. 
In  what  month  in  the  year  of  the  letter  was  tne  17th  a 
Thursday  ?  Such  are  the  problems  we  can  solve  through  a 
proper  understanding  of  the  calendar. 

The  first  difficulty  that  arises  in  devising  a  calendar 
comes  from  the  odd  lengths  of  the  week  and  the  year.  We 
all  know  that  there  are  seven  days  in  the  week,  and  we  have 
learned  that  the  year  contains  about  365£  days.  And  it  is 
impossible  to  divide  365  J  by  7  exactly,  without  a  "  re- 
mainder." Therefore  the  number  of  weeks  in  a  year  cannot 
be  expressed  as  a  whole  number ;  this  fact  makes  the  year 
and  the  week  " incommensurable,"  as  it  is  called.  The 
difficulty  could  not  be  avoided  by  changing  the  number  of 
days  in  the  week,  because  no  whole  number  of  days,  such 
as  6  or  9,  can  be  an  exact  divisor  of  365J. 

To  bring  about  an  exact  division,  it  would  be  necessary 

139 


ASTRONOMY 

to  change  either  the  length  of  the  day  or  the  length  of  the 
year.  But  neither  of  these  can  possibly  be  altered,  because 
both  are  natural  units  of  time.  The  day  (p.  66)  is  the 
quantity  of  time  required  by  the  earth  to  make  one  complete 
rotation  on  its  axis.  This  quantity  of  time  is  fixed  by  nature, 
and  is  therefore  called  a  natural  unit.  We  have  also  arti- 
ficial or  conventional  time-units,  such  as  the  hour  and  minute. 
For  instance,  the  hour  is  defined  conventionally  as  one 
twenty-fourth  part  of  the  time  required  by  the  earth  to 
complete  one  axial  rotation.  Being  an  artificial  unit,  it 
would  be  within  our  power  to  make  the  hour  one  twenty- 
fifth  or  one  twenty-third  part,  and  to  have  twenty-five  or 
twenty-three  hours  in  the  day.  This  makes  clear  the  dif- 
ference between  an  artificial  and  a  natural  unit  of  measure- 
ment :  one  is  man's  creation,  and  subject  to  change  by  him 
at  will ;  the  other  is  fixed  and  unchangeable  by  nature. 

But  chronology  does  not  concern  itself  with  minor  sub- 
divisions of  time,  such  as  hours  and  minutes ;  and  the  year 
of  chronology,  like  the  day,  is  a  natural  unit  quite  beyond 
our  control.  So  we  must  perforce  deal  with  the  year  and 
day  as  we  find  them;  our  artificial  chronological  units  are 
the  week  and  month.  We  have  just  seen  that  nothing  would 
be  gained  by  changing  the  number  of  days  in  the  week; 
we  may  add  that  it  would  be  impossible  practically  to  make 
such  a  change,  even  if  it  were  desirable.  Both  the  week 
and  the  month  have  acquired,  from  their  antiquity,  a  species 
of  historic  changelessness  which  lends  them  a  kind  of  per- 
manence almost  as  great  as  that  possessed  by  the  natural 
units  themselves. 

We  must  next  explain  what  is  meant  by  the  year  in 
chronology.  We  have  already  had  definitions  (p.  128) 
of  two  different  kinds  of  astronomic  years.  In  chronology, 

140 


THE   CALENDAR 

we  use  one  only  of  these  two  time-periods,  the  tropical  year. 
This  is  the  interval  of  time  between  two  dates  when  the 
sun,  in  its  apparent  motion  around  the  ecliptic  circle,  attains 
its  greatest  angular  distance  from  the  celestial  equator. 
When  this  occurs  at  the  summer  solstice  (p.  93)  we  have 
the  date  when  the  sun  climbs  highest  in  the  sky  at 
noon,  when  shadows  are  shortest,  when  midsummer  day 
occurs. 

These  facts  make  plain  at  once  the  reason  for  using  the 
tropical  year  in  calendar  making.  Suppose  we  have  become 
accustomed  to  midsummer  day  occurring  on  June  21. 
It  is  obvious  that  midsummer  must  necessarily  happen 
when  the  noon  shadows  are  shortest,  etc.  Now  suppose 
(to  exaggerate)  that  the  calendar  year  differed  by  a  day 
from  the  tropical  year.  If  one  midsummer  day  then  fell 
on  the  calendar  date  June  21,  the  next  midsummer  day 
would  fall  on  June  22.  And  each  midsummer  day  would 
come  a  day  later  in  its  turn,  until,  after  the  lapse  of  a 
century  or  so,  we  should  have  midsummer  in  December, 
and  our  calendar  would  be  completely  reversed.  The  one 
absolutely  essential  thing  is  to  have  the  calendar  year  as 
nearly  equal  to  the  tropical  year  as  it  is  possible  to 
make  it. 

We  have  seen  that  the  length  of  the  tropical  year  can  be 
determined  easily  by  astronomical  observations.  It  has 
been  found  to  contain  365.2422  days;  or,  approximately, 
365  J.  Now  the  calendar  year  must  of  course  contain  a 
round  number  of  days,  without  fractions ;  the  most  obvious 
way  to  bring  this  about  is  to  use  a  year  of  365  days,  and 
put  in  a  leap-year  of  366  days  every  fourth  year.  This  is 
the  Julian  calendar  already  mentioned  as  having  been  put 
in  force  under  Julius  Caesar. 

141 


ASTRONOMY 

The  error  of  this  calendar  is  found  easily  as  follows : 

JULIAN  CALENDAR 

1st  year  365  days 
2d  year  365  days 
3d  year  365  days 
4th  year  366  days 
Total,  4  years,  1461  days 

Actual  length  of  4  tropical  years  (365.2422  X  4)     1460.9688  days 
Error  of  Julian  calendar  .0312  day  in  4  years 

or     .0078  day  in  1  year 

The  above  simple  calculation  shows  that  the  Julian 
calendar  runs  into  error  at  the  rate  of  0.0078  day  per 
annum.  This  amounts  to  one  day  in  128  years,  and  the 
Julian  calendar  will  therefore  pass  out  of  accord  with  the 
true  tropical  motion  of  the  sun  at  that  rate. 

Another  simple  calculation  shows  how  to  correct  this 
error  almost  exactly,  and  this  leads  to  our  present  Gregorian 
calendar.  It  is  clear  that  the  Julian  method  of  introducing 
a  leap-year  every  four  years  somewhat  over-corrects  the 
error  that  would  be  caused  by  the  use  of  a  uniform  year  of 
365  days.  We  need  to  omit  one  of  those  leap-years  every 
128  years.  To  do  this  most  simply,  it  was  decided  under 
Pope  Gregory  to  omit  a  leap-year  once  every  century  for 
three  centuries;  and  in  every  fourth  century  to  omit  no 
leap-year.  This  omits  three  leap-years  in  400  years,  or 
one  in  133  years,  instead  of  one  in  128  years,  as  required. 
And  the  Gregorian  rule  for  leap-year  then  becomes  the 
following : 

The  year  is  a  leap-year  if  the  year  number  is  divisible 
exactly  by  4,  without  a  remainder ;  except  that  in  the  case 
of  century  years  like  1500,  1600,  etc.,  the  divisor  must  be 
400  instead  of  4. 

142 


THE   CALENDAR 

Under  this  rule  1912  was  a  leap-year ;  1900  was  not ;  but 
2000  will  be. 

Let  us  now  calculate  the  error  of  the  Gregorian  calendar. 
In  400  years  there  are  100  leap-years  in  the  Julian  calendar. 
The  exception  in  the  Gregorian  rule  reduces  this  number 
of  leap-years  to  97.  We  therefore  have  the  following  cal- 
culation : 

GREGORIAN  CALENDAR 

Number  of  days  in  400  years  (400  X  365)  is  146,000 
and  97  leap-year  days  97 

Total  number  of  days  in  400  calendar  years  146,097 

Number  of  days  in  400  tropical  years  (365.2422  X  400)  146,096.88 
Error  of  Gregorian  calendar  in  400  years  .12 

This  makes  the  Gregorian  error  in  one  year  only  .0003 
day;  so  that  3333  years  will  pass  before  there  is  an  accu- 
mulated total  error  of  a  single  day.  This  is  an  entirely 
negligible  quantity,  and  so  the  Gregorian  calendar  may  be 
regarded  as  perfectly  satisfactory  for  all  practical  purposes. 
Having  thus  explained  the  construction  of  the  calendar, 
the  next  step  is  to  show  how  to  calculate  the  week-day 
corresponding  to  any  date  in  the  past  or  future.  Let  us, 
for  convenience,  attach  to  the  seven  days  of  the  week  seven 
numbers,  thus : 

Sunday,  1, 

Monday,  2, 

Tuesday,  3, 

Wednesday,  4, 

Thursday,  5, 

Friday,  6, 

Saturday,  7. 

Let  us  also  designate  as  the  "  century  number "  the  first 
two  digits  of  the  year  number.     Thus,  in  1913,  19  is  the 

143 


ASTRONOMY 

century  number  and  1913  is  the  year  number.     Then  we 
have  the  following : 1 

RULE  FOR  FINDING  THE  WEEK-DAY 

1.  Divide  the  century  number  by  4  and  7,  and  call  the 
remainders  resulting  from  the  division  the  first  and  second 
remainders. 

2.  Divide  the  year  number  by  4  and  7,  and  call  the  re- 
mainders the  third  and  fourth  remainders. 

3.  Add  five  times  the  first  remainder  to  the  second  re- 
mainder, and  call  the  sum  the  "  const  ant. " 

4.  Add  the  following  five  numbers ;    viz. :    the  constant ; 
five  times  the  third  remainder;    three  times  the  fourth  re- 
mainder ;  the  day  of  the  month ;  and  the  following  number 
depending  on  the  name  of  the  month : 

in  Jan. ;  6,  in  ordinary  years ;  5,  in  leap-years ; 
"  Feb. ;  2,  in  ordinary  years ;  1,  in  leap-years ; 
"  March;  2,  in  all  years; 


"April; 

5,  " 

n 

"  May  ; 

o," 

11 

"  June  ; 

3,  " 

(i 

"July; 

5,  " 

ft' 

"Aug.; 

1,  " 

tt 

"Sept.; 

4,  " 

" 

"  Oct.  ; 

6," 

ft 

"  Nov.  ; 

2,  " 

tt 

"Dec.; 

4,  " 

tt 

And  call  the  sum  of  the  five  numbers  thus  added  the  "sum." 
5.  Divide  this  sum  by  7,  and  call  the  remainder  the  fifth 

remainder. 

Then  this  fifth  remainder,  when  increased  by  unity,  will 

be  the  week-day  number  required. 

1  For  a  demonstration  of  this  rule,  see  Note  15,  Appendix. 

144 


THE   CALENDAR 

As  an  example,  let  us  find  the  week-day  corresponding 
to  July  4,  1913.  We  have : 

1.  19  divided  by  4  gives  first  remainder,  3 ; 
19  divided  by  7  gives  second  remainder,  5. 

2.  1913  divided  by  4  gives  third  remainder,  1 ; 
1913  divided  by  7  gives  fourth  remainder,  2. 

3.  Five  times  first  remainder,  15, 

second  remainder,  5, 

The  constant  is  20. 

4.  The  constant,  20, 
Five  times  the  third  remainder,  5, 
Three  times  the  fourth  remainder,  6, 
Day  of  month,  July  4,  4, 
The  month  number  for  July,  5, 
The  sum  is  40. 

5.  40  divided  by  7  gives  fifth  remainder,     5. 

The  fifth  remainder  increased  by  unity  gives  the  week-day 
number  as  6,  corresponding  to  Friday.  Therefore  July  4, 
1913,  is  a  Friday. 

The  above  rule  applies  to  the  Gregorian  calendar;  but 
we  may  use  it  in  the  Julian  calendar  also  if  we  simply  omit 
the  first  and  second  remainders,  and  for  the  constant  always 
use  0. 

The  foregoing  method  of  calculation  may  be  replaced  by 
a  device  called  a  Perpetual  Calendar,  by  means  of  which 
all  calendar  problems  may  be  solved  with  ease.  The  ac- 
companying form  of  perpetual  calendar  was  arranged  by 
Captain  John  Herschel ;  it  is  convenient  in  use,  and  may  be 
extended  easily,  indefinitely  in  either  direction,  backwards 
from  1860,  or  forwards  from  1995,  for  which  limiting  dates 
it  is  here  given.  The  months  January  and  February  appear 
twice  in  it ;  the  italicized  January  and  February  to  be  used 
in  leap-years,  which  are  also  italicized  in  the  columns  of 
year  numbers.  In  ordinary  years  the  unitalicized  January 
L  145 


ASTRONOMY 

and  February  are  to  be  used.  The  calendar  is  Gregorian. 
The  following  examples  will  illustrate  the  use  of  this  per- 
petual calendar  in  finding  the  fourth  constituent  part  of  a 
date  for  which  three  parts  are  given. 

1.  What  day  of  the  week  is  July  4,  1913?     Opposite  4, 
under  Day  of  the  Month,  and  in  the  column  headed  July, 
we  find  the  letter  F.    We  then  find  1913  in  the  third  ver- 
tical column  of  year  numbers.     Running  up  this  column  to 
the  letter  F,  and  thence  turning  to  the  right,  we  find  Friday 
for  the  day  of  the  week.     This  agrees  with  our  former  cal- 
culation by  the  rule. 

2.  In  what  years  following  leap-years  does  inauguration 
day,  March  4,  fall  on  a  Sunday?     Opposite  4,  and  under 
March,  we  find  the  letter  B.     Opposite  Sunday,  B  occurs 
in  the  first  column.     Consequently,   March  4  is  Sunday  in 
1860,   1866,   1877,   1883,  etc.     But  the  only  years  in  this 
column  that  follow  leap-years  are  1877,  1917,  1945,  and  1973. 
In  these  years,  therefore,  inauguration  day  falls  on  Sunday. 

Having  thus  explained  the  civil  calendar  in  ordinary  use, 
we  shall  next,  to  complete  the  subject,  describe  the  Ecclesi- 
astical Calendar  as  briefly  as  possible.  The  fundamental 
problem  of  this  calendar  is  to  find  the  date  of  Easter  Sunday 
in  any  given  year.  Following  Gauss,1  we  shall  divest  the 
subject  of  all  non-essential  details ;  and  especially  exclude 
the  ancient  terminology,  which  tends  to  involve  this  some- 
what complicated  problem  in  unnecessary  obscurity.  . 

Fundamental  data  are  to  be  found  in  regulations  adopted 
by  the  famous  Ecclesiastical  Council  of  Nice,  which  met  in 
the  year  325  A.D.  According  to  decree  of  that  council, 
Easter  Sunday  is  the  first  Sunday  that  follows  the  first 

1  Gauss,  Berechnung  des  Osterfestes;  v.  Zach's  Monatliche  Correspondenz, 
August  1800. 

146 


THE   CALENDAR 


PERPETUAL  CALENDAR 


DAY  OF  THE 
MONTH 

JAN. 
OCT. 

APR. 
JULY 
Jan. 

SEPT. 
DEC. 

JUNE 

FEB. 
MAR. 
Nov. 

AUG. 

Feb. 

MAY 

1  8  15  22  29 

A 

B 

C 

D 

E 

F 

G 

Monday 

2  9  16  23  30 

G 

A 

B 

C 

D 

E 

F 

Tuesday 

3  10  17  24  31 

F 

G 

A 

B 

C 

D 

E 

Wednesday 

4  11  18  25 

E 

F 

G 

A 

B 

C 

D 

Thursday 

5  12  19  26 

D 

E 

F 

G 

A 

B 

C 

Friday 

6  13  20  27 

C 

D 

E 

F 

G 

A 

B 

Saturday 

7  14  21  28 

B 

C 

D 

E 

F 

G 

A  ' 

Sunday 

1860 

1861 

1862 

1863 

1864 

1865 

1866 

1867 

1868 

1869 

1870 

1871 

What  day 

1872 

1873 

1874 

1875 

1876 

of  the  week 

1877 

1878 

1879 

1880 

1881 

1882 

was 

1883 

1884 

1885 

1886 

1887 

March  4, 

1888 

1889 

1890 

1891 

1892 

1893 

1865? 

1894 

1895 

1896 

1897 

1898 

1899 

1900 

1901 

1902 

1903 

1904 

1905 

Under 

1906 

1907 

1908 

1909 

1910 

1911 

March,  op- 

1912 

1913 

1914 

1915 

1916 

posite  4  is 

1917 

1918 

1919 

1920 

1921 

1922 

the  letter 

1923 

1924 

1925 

1926 

1927 

B.  In  the 

1928 

1929 

1930 

1931 

1932 

1933 

1865   col- 

1934 

1935 

1936 

1937 

1938 

1939 

umn,  oppo- 

1940 

1941 

1942 

1943 

1944 

site  B,  is 

1945 

1946 

1947 

1948 

1949 

1950 

Saturday. 

1951 

1952 

1953 

1954 

1955 

1956 

1957 

1958 

1959 

1960 

1961 

Note   In 

1962 

1963 
1968 

1969 

1964 
1970 

1965 
1971 

1966 

1967 
1972 

leap-years, 

T 

1973 

1974 

1975 

1976 

1977 

1978 

and   Feb. 

1979 

1980 

1981 

1982 

1983 

that  are  un- 

1984 

1985 

1986 

1987 

1988 

1989 

derlined. 

1990 

1991 

1992 

1993 

1994 

1995 

147 


ASTRONOMY 

full  moon  occurring;  on  or  after  March  21,  the  date  of  the 
vernal  equinox^  (p.  72).  And  it  was  further  ordered  that 
the  day  of  the  ecclesiastical  full  moon  shall  be  the  fourteenth 
day  after  new  moon,  or  the  day  when  the  so-called  "  moon's 
age"  is  14  days.  Our  problem  is  to  calculate  the  date  of 
Easter  Sunday  in  accordance  with  this  regulation. 

Gauss'  rule  l  is  approximately  as  follows,  some  of  the 
successive  operations  being  identical  with  those  already  used 
for  the  ordinary  calendar : 

1.  Divide  the  century  number  by  4  and  7,  and  call  the 
remainders*  resulting  from  the  division  the  first  and  second 
remainders. 

2.  Divide  the  year  number  by  4  and  7,  and  call  the  re- 
mainders the  third  and  fourth  remainders. 

3.  Divide  the  year  number  by  19,  and  call  the  remainder 
the  sixth  remainder. 

4.  Add  five  times  the  first  remainder  to  the  second  re- 
mainder, and  call  the  sum  the  "  constant." 

5.  Add  eight  times  the   century  number  to  the   number 
13  ;  divide  the  sum  by  25,  and  call  the  remainder  the  eighth 
remainder. 

6.  Subtract  the  first  remainder  from  the  century  number, 
divide  the  difference  by  4,  and  call  the  quotient  (which  will 
be  a  whole  number)  the  result  of  operation  6. 

7.  Add  eight  times  the  century  number  to  the  number  13, 
deduct  from  the  sum  the  eighth  remainder,  divide  what  is 
left  by  25,  and  call  the  quotient  (which  will  be  a  whole 
number)  the  result  of  operation  7. 

8.  Add  the  century  number  to  the  number  15,  deduct 
from  the  sum  the  results  of  operations  6  and  7,  and  call 
what  is  left  the  result  of  operation  8. 

1  For  a  demonstration,  see  Note  16,  Appendix. 
148 


THE   CALENDAR 

9.  Add  the  result  of  operation  8  to  19  times  the  sixth 
remainder,  divide  by  30,  and  call  the  remainder  resulting 
from  the  division  the  seventh  remainder. 

10.  Add  five  times  the  third  remainder,  three  times  the 
fourth  remainder,   the   constant,   the   number   2,   and  the 
seventh  remainder;    divide  by  7,  and  call  the  resulting  re- 
mainder the  ninth  remainder. 

11.  Subtract  the  ninth  remainder  from  the  seventh  re- 
mainder, and  increase  the  difference  by  28.     The  result  will 
be  the  date  of  Easter  Sunday  in  March.     But  if  this  date 
is  greater  than  31,  Easter  Sunday  will  fall  in  April,  and  its 
date  in  April  will  be  found  by  subtracting  the  ninth  re- 
mainder from  the  seventh,  as  before,  and  diminishing  the 
difference  by  3. 

The  above  rule  applies  to  the  Gregorian  calendar.  In 
the  Julian  calendar  the  same  rule  may  be  used ;  but  the 
constant  is  then  always  0,  and  the  result  of  operation  8 
always  15.  Furthermore,  two  exceptions  to  the  above  rule 
exist  in  the  Gregorian  calendar : 

1.  When  Easter  Sunday  comes  on  April  26  by  the  rule, 
April  19  must  be  substituted  for  April  26. 

2.  Take  eleven  times  the  result  of  operation  8,  and  in- 
crease the  product  by  the  number  11.     Divide  the  sum  by 
30.     If  the  remainder  resulting  from  this  division  is  less 
than  19,  and  if  at  the  same  time  the  seventh  remainder  was 
28  and  the  ninth  remainder  0,  the  rule  will  give  April  25 
for  the  date  of  Easter  Sunday.     When  these  conditions  all 
occur,  substitute  April  18  for  April  25. 


149 


ASTRONOMY 

As  an  example,  let  us  calculate  the  date  of  Easter  Sunday 
in  1913.     We  have  : 

1.  Century  number  19  divided  by  4,  gives  first  remainder  3, 
Century  number  19  divided  by  7,  gives  second  remainder  5. 

2.  Year  number  1913  divided  by  4,  gives  third  remainder  1, 
Year  number  1913  divided  by  7,  gives  fourth  remainder  2. 

3.  Year  number  1913  divided  by  19,  gives  sixth  remainder  13. 

4.  The  constant  is  20. 

5.  The  eighth  remainder  is  15. 

6.  Result  of  operation  6  is  4. 

7.  Result  of  operation  7  is  6. 

8.  Result  of  operation  8  is  24. 

9.  The  seventh  remainder  is  1. 

10.  The  ninth  remainder  is  6. 

11.  Easter  Sunday  is  March  (1-6  -f  28),  or  March  23. 


150 


CHAPTER  IX 

NAVIGATION 

THE  guiding  of  a  ship  across  the  unmarked  trackless 
ocean  is  strictly  a  problem  of  astronomy :  among  the  many 
problems  of  the  science  it  is  the  most  important  commer- 
cially ;  certainly  there  is  no  other  so  astonishing  to  those 
who  do  not  understand  the  simple  methods  employed  for 
its  solution.  Briefly  stated,  the  astronomic  problem  of 
navigation  consists  in  ascertaining  a  ship's  latitude  and 
longitude  by  observing  the  heavenly  bodies.  If  this  can 
be  done,  we  know  the  exact  position  of  the  ship  on  the 
earth's  surface ;  and  knowing  also  the  latitude  and  longi- 
tude of  the  port  to  which  the  ship  is  bound,  it  becomes  an 
easy  matter  to  calculate,  or  to  measure  on  a  chart,  whether 
the  ship  must  be  steered  north,  east,  south,  or  west,  in 
order  to  reach  its  destination. 

Inasmuch  as  ocean  currents,  leeway,  or  other  causes  may 
produce  unperceived  deflections  as  a  ship  moves  through 
the  water,  it  is  necessary  and  customary  for  navigators  to 
determine  their  position  astronomically  at  frequent  inter- 
vals; once  each  day,  if  possible.  These  successive  astro- 
nomical observations  furnish  a  continuous  check  upon  the 
running  of  the  ship.  Each  new  observation  gives  a  new 
11  departure,"  as  it  is  called;  and  helps  to  assure  a  correct 
" land-fall"  at  the  end  of  the  voyage,  as  sailors  say.  When 
clouds  prevent  satisfactory  observations  of  the  sky,  the 
ship  must  be  run  by  "dead  reckoning" ;  a  very  expressive 

151 


ASTRONOMY 

term  which  indicates  how  little  confidence  mariners  have  in 
their  compass,  as  compared  with  observations  of  the  un- 
erring heavens. 

Our  concern  is  with  the  purely  astronomic  question  of 
navigation ;  with  seamanship,  knotting  and  splicing,  charts 
and  compass,  leadline  and  sounding  machine,  we  have 
nothing  to  do.  And  the  astronomic  problem  is  a  very 
simple  one,  for  both  the  latitude  and  longitude  of  the  ship 
may  be  calculated  if  we  measure  with  a  suitable  instrument, 
and  in  a  suitable  way,  the  angular  elevation  or  altitude 
(p.  36)  of  the  sun  above  the  visible  sea-horizon. 

The  instrument  used  for  this  purpose  at  sea  is  called  a 
Sextant ;  it  may  be  defined  as  a  portable  instrument  for 
measuring  the  angular  altitude  of  the  sun  above  the  horizon. 
Figure  38  will  enable  the  reader  to  form  an  idea  of  its  ap- 
pearance, and  to  understand  its  principle.  The  essential 
parts  are  two  small  silvered  mirrors,  M  and  m ;  a  telescope, 
EK]  and  a  circle,  A  A,  engraved  with  "  graduations "  as 
they  are  called,  by  means  of  which  angles  may  be  measured 
upon  it  in  degrees,  minutes,  and  seconds.  The  mirror  m 
and  the  telescope  EK  are  firmly  attached  to  the  sextant ; 
but  the  mirror  M  is  pivoted  in  such  a  way  that  it  can  be 
turned,  and  the  angle  through  which  it  is  turned  measured 
on  the  circle  by  means  of  the  index  CB.  When  the  mirror 
M  is  turned  back  until  it  is  parallel  to  the  mirror  m,  the 
circle  reads  0°,  because  the  angle  between  the  two  mirrors 
is  then  0°.  In  all  other  positions  the  circle  measures  the 
angle  between  the  two  mirrors.  P  and  Q  are  sets  of  colored 
glasses,  which  can  be  interposed  temporarily,  when  the 
sun's  rays  are  so  brilliant  as  to  be  hurtful  to  the  observer's 
eye.  R  is  a  small  magnifying  glass,  pivoted  at  S,  intended 
to  facilitate  the  examination  of  the  index  CB.  At  C  and  B 

152 


NAVIGATION 

are  shown  the  "  clamp "  by  means  of  which  the  index  can 
be  fastened  to  the  circle,  and  the  " tangent-screw,"  which 
will  adjust  it  delicately,  after  it  has  been  "  clamped."  /  and 
F  are  accessories  for  the  telescope. 

The  mirror  m  has  an  important  peculiarity.  The  silvering 
is  scraped  away  at  the  back  of  the  mirror  from  one-half  its 
surface.  Thus  only  one-half  reflects;  the  other  half  is 


FIG.  38.    The  Sextant. 
(From  Bowditch's  Navigator,  1912  ed.,  p.  66,  Bureau  of  Navigation,  U.  S.  Navy.) 

simply  transparent  glass.  A  navigator  looking  into  the 
telescope  at  E  will  therefore  look  through  the  mirror  with 
half  his  telescope,  and  with  the  other  half  he  will  look  into 
the  mirror. 

Now  it  is  a  fact  that  half  a  telescope  acts  just  like  a  whole 
one,  always.  If  a  person  using  an  ordinary  " spy-glass" 
covers  half  of  the  big  end  with  his  hand,  he  will  see  the  same 
view  he  saw  with  the  whole  glass.  Only,  as  half  the  light- 

153 


ASTRONOMY 

gathering  power  is  cut  off,  this  view  will  be  fainter  or  dim- 
mer, —  less  luminous.  Applying  this  fact  to  the  sextant 
telescope,  it  is  clear  that  the  observer  will  see  two  things 
at  once  with  the  telescope :  he  will  see  what  is  visible 
through  the  mirror  ra  with  half  the  telescope;  and  with 
the  other  half  he  will  see  what  is  visible  by  reflection  from 
the  mirror  m. 

If  he  holds  the  sextant  in  such  a  position  that  the  telescope 
is  horizontal,  he  will  see  the  visible  sea-horizon  with  half 
the  telescope  through  the  mirror.  If  the  other  mirror  M 
is  then  turned  to  the  proper  position,  and  the  sextant  held 
in  the  hand  with  its  telescope  still  horizontal,  and  its  circle 
vertical,  it  is  possible  to  see  the  sun  at  the  same  time  with 
the  other  half  of  the  telescope,  the  solar  rays  having  been 
reflected  from  both  mirrors.  To  make  this  possible,  the 
horizontal  telescope  must,  of  course,  be  aimed  at  that  point 
of  the  sea-horizon  which  is  directly  under  the  sun.  The 
solar  rays  will  then  strike  the  mirror  M  first;  be  thence 
reflected  to  the  silvered  part  of  the  mirror  m ;  and  finally 
into  the  telescope.  So  the  observation  consists  in  so  adjust- 
ing or  turning  the  mirror  M ,  that  the  sun  and  the  horizon 
can  be  seen  coincidently  in  the  telescope. 

The  angle  between  the  mirrors  can  then  be  measured  on 
the  circle ;  and  it  is  easy  to  prove  l  that  the  angular  alti- 
tude of  the  sun  will  be  twice  the  angle  between  the  two 
mirrors.  Thus  the  sextant  becomes  an  instrument  for 
measuring  the  sun's  altitude;  it  remains  to  explain  how  a 
knowledge  of  that  altitude  will  furnish  us  with  the  ship's 
latitude  and  longitude. 

To  obtain  the  ship's  latitude,  it  is  best  to  measure  the 
solar  altitude  when  the  sun  is  on  the  meridian,  at  apparent 

1  Note  17,  Appendix. 
154 


NAVIGATION 

solar  noon.  Omitting  certain  very  small  corrections,  the 
sun  will  then  have  its  greatest  altitude  for  the  day ;  so  that 
the  navigator  need  only  begin  measuring  altitudes  a  few 
minutes  before  noon,  and  continue  as  long  as  the  altitude 
is  increasing.  The  moment  it  begins  to  diminish,  he  stops 
and  "reads"  his  sextant  circle;  thus  obtaining  the  meridian 
altitude  of  the  sun.  The  accompanying  Fig.  39  shows  how 
the  latitude  is  obtained  from  such  an  observation.  0  is 
the  observer  on  the  ship.  The  semicircle  H'PSEH  is  that 
half  of  the  celestial  meridian  (p.  36)  which  is  above  the  hori- 
zon. H  and  Hf  are  the  south  and  north  points  of  the  hori- 
zon, where  it  is  intersected 
by  the  celestial  meridian. 
P  is  the  north  celestial 
pole  (p.  31),  and  S  the 
sun  as  observed  on  the 
celestial  meridian.  E,  90° 
from  the  pole,  is  a  point  h  °  M 

FIG.  39.    Latitude  from  Observation. 

on   the   celestial    equator, 

where  it  crosses  the  meridian.  The  angle  SOH,  or  the  arc 
SH ,  is  then  the  observed  altitude  of  the  sun ;  and  the  arc 
SE  is  the  declination  (p.  34)  of  the  sun.  This  declination 
is  always  known ;  it  can  be  calculated  in  advance,  because 
we  know  the  annual  orbit  of  the  earth  around  the  sun  and 
the  point  of  the  ecliptic  circle  (p.  27)  at  which  the  sun 
appears  on  the  date  when  the  observation  was  made.  In 
fact,  the  navigator  always  has  at  hand  a  copy  of  the 
Nautical  Almanac,  which  is  a  book  published  annually  by 
the  United  States  government,  in  which  the  sun's  declina- 
tion is  printed  for  every  day  in  the  year. 

The  navigator  then  simply  subtracts  the  known  declina- 
tion SE  from  the  observed  altitude  SH,  and  thus  obtains 

155 


ASTRONOMY 

the  arc  EH,  or  the  altitude  of  the  celestial  equator  above' 
the  south  point  of  the  horizon.  As  soon  as  EH  becomes 
known,  it  is  easy  to  obtain  the  latitude.  For  the  arc  PE 
is  also  known;  it  is  always  90°,  because  it  is  the  angular 
distance  from  the  equator  to  the  pole.  Therefore  we  need 
merely  subtract  PE  and  EH  from  180°,  to  get  PH' ,  the 
angular  altitude  of  the  celestial  pole  above  the  horizon. 
But  this  (p.  40)  is  always  equal  to  the  latitude ;  and  so  the 
latitude  of  the  ship  becomes  known  from  the  sextant  measure- 
ment of  altitude. 

In  making  observations  of  this  kind,  it  is  necessary  to 
apply  certain  corrections  to  the  observed  altitude,  of  which 
the  two  most  important  are  the  correction  for  refraction, 
and  the  correction  for  semi-diameter.  The  former  is  due 
to  the  bending  of  the  sun's  light  as  it  comes  down  to  us 
through  the  terrestrial  atmosphere  (p.  114).  The  amount 
of  this  bending  can  be  found  in  refraction  tables  which  are 
printed  in  all  books  on  navigation ;  the  navigator  merely 
subtracts  it  from  the  altitude  as  actually  observed. 

The  other  correction  for  semi-diameter  is  due  to  the 
fact  that  we  cannot  measure  the  altitude  of  the  sun's  center 
because  the  sun  appears  in  the  sextant  telescope  as  a  round 
disk,  and  it  is  impossible  to  estimate  the  position  of  its 
center  accurately.  Therefore  navigators  always  measure 
from  the  lowest  or  highest  point  of  the  disk :  in  either  case, 
the  angular  semi-diameter  must  be  added  to  or  subtracted 
from  the  observed  altitude  to  get  the  altitude  of  the  sun's 
center.  This  semi-diameter  varies  a  little  through  the  year 
because,  as  we  know,  the  flattening  of  the  earth's  orbit 
around  the  sun  (p.  118)  alternately  increases  and  diminishes 
the  distance  of  the  earth  from  the  sun.  But  the  exact 
value  of  the  semi-diameter  is  printed  for  each  day  in  the 

156 


NAVIGATION 

nautical  almanac,  whence  the  navigator  obtains  it,  together 
with  the  sun's  declination. 

To  ascertain  the  ship's  longitude,  a  somewhat  different 
process  is  employed.  In  principle  it  depends  upon  the  time- 
differences  which,  as  we  have  seen,  exist  between  different 
places  on  the  earth  (p.  72).  Strictly  speaking,  longitudes 
are  longitude  differences.  The  longitude  of  New  York,  for 
instance,  is  really  the  longitude  difference  of  New  York 
from  Greenwich.  And  the  time  difference  between  New 
York  and  Greenwich  corresponds  exactly  to  the  longitude 
difference,  one  hour  of  time  corresponding  to  each  fifteen 
degrees  of  longitude. 

The  navigator  takes  advantage  of  these  facts  in  a  very 
simple  way.  He  carries  in  the  ship  one  or  more  marine 
chronometers.  These  are  merely  very  large  watches,  accu- 
rately made,  and  mounted  in  boxes  with  swinging  supports, 
so  as  to  prevent  the  ship's  rolling  from  influencing  the  exact 
running  of  the  instrument.  Before  leaving  port,  these 
ship's  chronometers  are  " rated"  accurately,  by  comparing 
them  on  successive  days  with  standard  telegraphic  time 
signals  from  some  astronomical  observatory. 

By  rating  a  chronometer  we  do  not  mean  merely  ascer- 
taining its  error,  or  the  number  of  minutes  and  seconds  it 
may  be  fast  or  slow  on  a  given  date.  Rating  includes  also 
the  determination  of  the  fraction  of  a  second  by  which  the 
chronometer  increases  or  diminishes  its  error  on  each  suc- 
ceeding day.  For  instance,  if  a  chronometer  is  found  to 
have  the  following  error  and  rate : 

April  15,  1913,  chronometer  fast  28.0  seconds,  and  gaining  0.3  daily ; 

then  on  April  30,  1913,  fifteen  days  later,  the  chronometer 
would  be  fast  28.0  +  15  X  0.3  seconds,  or  32.5  seconds. 

157 


ASTRONOMY 

Knowing  the  error  and  rate,  the  navigator  can  always  ob- 
tain correct  time  from  his  chronometers,  within  the  limits 
of  accuracy  with  which  they  can  be  made  to  maintain  a 
constant  or  unvarying  rate. 

Marine  chronometers  are  always  set  to  Greenwich  time ; 
so  that  when  a  navigator  takes  the  time  from  the  chro- 
nometer, allowing  for  its  rate,  it  is  always  Greenwich  time. 
Now  suitable  sextant  observations  enable  him  to  determine 
also  the  correct  local  mean  solar  time  of  the  ship;  this 
having  been  done,  a  simple  comparison  with  the  Greenwich 
time  of  the  chronometer  furnishes  the  time  difference  between 
the  ship  and  Greenwich,  and  therefore  also  the  longitude 
difference,  or  "  longitude  of  the  ship." 

It  remains  to  explain  how  the  ship's  local  time  may  be 
ascertained  by  observation  with  the  sextant.  This  is  ac- 
complished by  observing  the  altitude  of  the  sun,  just  as  we 
have  explained  in  the  case  of  latitude  determinations; 
only,  while  latitude  observations  are  made  at  noon,  time 
or  longitude  observations  must  be  made  rather  early  in  the 
morning,  or  late  in  the  afternoon. 

It  is  quite  obvious  that  a  measurement  of  the  altitude  or 
angular  elevation  of  the  sun  in  the  sky  must  make  the 
time  of  day  known.  For  the  altitude  is  zero  at  sunrise, 
and  greatest  at  noon ;  consequently,  if  we  know  the  alti- 
tude, we  must  be  able  to  calculate  1  how  far  the  sun  has 
proceeded  from  sunrise  toward  noon  in  its  apparent  diurnal 
rotation  across  the  sky.  The  calculation  involves  the  use 
of  spherical  trigonometry,  and  cannot  be  explained  in  detail 
here ;  but  enough  has  been  said  to  show  that  such  a  cal- 
culation is  possible. 

The  methods  given  here  for  navigating  a  ship  are  the 

1  Note  18,  Appendix.    —  ^Q  ^  ^ 
158 


NAVIGATION 

simplest  and  most  easily  understood.  Many  other  methods, 
or  modifications  of  the  above  methods,  have  been  devised, 
and  may  be  found  in  any  standard  book  on  navigation. 

Older  methods  are  perhaps  of  minor  interest,  but  the 
reader  will  surely  wish  to  know  how  ships  were  navigated 
before  the  days  of  chronometers.  The  first  chronometer 
capable  of  keeping  reasonably  accurate  time  at  sea  was  not 
made  until  1736,  although  it  was  in  1675  that  Charles  II 
issued  his  royal  warrant  establishing  the  office  of  astronomer 
royal,  and  making  it  the  duty  of  that  official  to  "  apply 
himself  with  the  most  exact  care  and  diligence  to  the  rectify- 
ing of  the  tables  of  the  motions  of  the  heavens  and  the 
places  of  the  fixed  stars,  in  order  to  find  out  the  so  much 
desired  longitude  at  sea,  for  the  perfecting  the  art  of  naviga- 
tion." 

Without  the  chronometer  the  navigator  could  still  obtain 
his  local  time,  but  he  had  no  Greenwich  time  with  which  to 
compare  it.  But  his  latitude  from  a  noon  observation  was 
always  available,  since  comparatively  rude  instruments  for 
measuring  altitudes  existed  for  centuries  before  the  inven- 
tion of  the  sextant.  Thus,  in  the  early  days,  the  navigator 
was  forced  to  find  his  way  with  latitudes  only.  For  in- 
stance, in  a  voyage  from  England  to  Rio,  the  ship  would 
be  steered  southward  and  westward,  more  or  less,  until  the 
"  noon-sights "  showed  that  the  latitude  of  Rio  had  been 
reached.  It  was  then  merely  necessary  to  steer  due  west, 
along  the  latitude  parallel  of  Rio,  and  checking  the  latitude 
of  the  ship  by  daily  noon-sights ;  the  lookout  man  forward 
would  notify  the  navigator  when  he  " raised  the  land." 
But  with  no  knowledge  of  longitude,  and  in  a  sailing  ship, 
the  navigator  might  be  uncertain  by  many  weeks  as  to  the 
date  when  he  would  reach  the  port  of  Rio. 

159 


CHAPTER  X 

MOONSHINE 

THE  moon,  more  than  any  other  celestial  body,  is  in  a 
very  peculiar  sense  our  own,  for  it  is  a  satellite  of  the  earth, 
revolving  around  us,  and  accompanying  our  annual  orbital 
journey  about  the  sun.  Earth  and  moon  together  follow 
the  same  path,  completing  each  year  a  full  circuit  around  the 
sun.  And  the  moon  is  important,  too,  in  the  history  of 
astronomy  :  upon  its  peculiarly  intricate  motions ;  its  con- 
nection with  eclipses;  its  lifting  of  the  great  waters  of 
ocean  in  tidal  ebb  and  flow, — upon  a  due  explanation  of  all 
these  things  men  have  exercised  their  highest  powers  from 
the  very  beginning  of  the  science. 

The  moon  is  not  self-luminous  like  the  sun,  but  shines 
only  by  receiving  light  from  the  sun,  and  reflecting  it  to  the 
earth  (cf.  p.  16).  Its  orbit  around  the  earth,  like  most 
orbits,  is  a  slightly  flattened  oval  or  ellipse,  with  the  earth 
situated  at  the  focus  (cf.  p.  116),  a  little  to  one  side  of  the 
center.  The  orbital  plane  can  be  imagined  extended  out- 
ward indefinitely,  so  as  to  cut  out  a  great  circle  on  the 
celestial  sphere,  similar  to  the  ecliptic  circle  (p.  27).  Some- 
where in  this  great  circle  the  moon  will  always  be  seen 
projected  on  the  sky.  The  plane  of  the  lunar  orbit  is  in- 
clined to  the  ecliptic  plane  by  a  small  angle,  about  5° ;  so  that 
this  is  also  the  angle  between  the  ecliptic  circle  on  the  celestial 
sphere  and  the  great  circle  belonging  to  the  lunar  orbit.1 

1  We  learn  in  Spherical  Geometry  that  the  angle  between  any  two 
great  circles  drawn  upon  a  sphere  is  equal  to  the  angle  between  the  two 
planes  in  which  the  circles  are  situated. 

160 


MOONSHINE 

As  the  moon  travels  around  the  earth  in  its  orbit,  we  see 
it  projected  on  the  sky,  and  apparently  progressing  around 
its  orbital  great  circle,  just  as  the  sun  appears  to  travel 
around  the  ecliptic  circle.  No  wonder  the  ancients  were 
puzzled  when  they  saw  both  sun  and  moon  alike  moving 
around  the  sky  in  their  respective  great  circles.  Of  course 
they  thought  both  bodies  were  alike  revolving  around  the 
earth,  and  concluded  that  the  earth  must  be  the  immobile 
center  of  all  things.  But  we  now  know  that  the  moon 
appears  to  progress  around  the  sky  because  it  is  really  mov- 
ing around  the  earth,  and  we  see  this  real  motion  projected 
on  the  sky.  The  sun,  on  the  other  hand,  only  seems  to 
travel  around  the  sky ;  it  is  really  stationary,  and  its  motion 
is  an  apparent  one,  due  to  the  real  motion  of  the  earth  in  its 
own  annual  orbit  (p.  116).  But  to  the  eye  both  sun  and 
moon  alike  seem  to  circle  the  sky. 

The  angular  velocity  of  lunar  motion  in  the  moon's  pro- 
jected great  circle  is  far  greater  than  that  of  the  sun  in  its 
ecliptic  circle.  Both  bodies  appear  to  move  in  the  same 
direction,  from  west  to  east ;  but  while  the  solar  apparent 
revolution  takes  about  a  year,  and  therefore  averages  about 
1°  daily,  the  moon  completes  a  circuit  from  any  fixed  star 
back  to  the  same  star  again  in  about  27 J  days,  correspond- 
ing to  an  average  daily  angular  motion  of  about  13°.  This 
period  of  27j  days,  from  star  to  star,  is  called  the  lunar 
^idereal  Period,  and  corresponds  to  the  sidereal  year  (p. 
128)  of  terrestrial  orbital  motion  around  the  sun.  But  the 
moon  has  also  another  period,  called  the  Synodic  Period. 
To  understand  it,  we  must  remember  that  as  the  moon's 
angular  motion  among  the  stars  is  about  thirteen  times  as 
rapid  as  the  sun's  apparent  angular  motion,  the  moon  must 
be  constantly  overtaking  and  passing  the  sun,  much  as  the 

M  161 


ASTRONOMY 

minute  hand  of  a  watch  is  constantly  overtaking  the  hour 
hand.  The  synodic  period  is  defined  as  the  interval  of  time 
between  two  such  successive  overtakings  of  the  sun  by  the 
moon.  It  is  about  29J  days  long,  or  about  2^  days  longer 
than  the  sidereal  period. 

To  explain  this  in  a  different  way,  suppose  the  moon  and 
the  sun  are  to-day  both  projected  on  the  sky  near  a  certain 
fixed  star.  Then,  27J  days  later,  the  moon  will  have  circled 
the  sky  completely,  and  will  be  back  near  the  same  star. 
During  the  27J  days,  however,  the  sun  will  have  moved 
apparently  some  27i°  eastward,  because  of  its  apparent 
motion  in  the  ecliptic  circle.  Therefore,  to  rejoin  the 
sun,  the  moon  will  still  need  to  travel  those  27|° ;  and 
this,  at  the  rate  of  about  13°  daily,  will  require  approxi- 
mately 2j  days.  So  the  synodic  period  is  again  seen  to  be 
29J  days  long. 

Probably  the  first  astronomical  phenomenon  ever  ob- 
served by  man  was  the  " waxing  and  waning"  of  the  moon; 
its  change  in  shape  from  a  thin  crescent,  gradual,  night 
after  night,  to  the  "  half  -moon "  of  Plate  3,  p.  17;  and 
finally  its  increase  to  the  brilliant  circular  orb  we  call  the 
full-moon.  The  accompanying  Plate  6  is  a  photograph  of 
the  moon,  nearly  full;  and  the  small  additional  picture  is 
the  crescent  moon.  The  dim  visibility  of  the  remaining 
lunar  surface  within  the  crescent  is  explained  on  p.  164. 

What  are  these  "  phases "  of  the  moon,  and  what  is  their 
cause?  We  have  just  seen  that  the  moon  is  not  self-lumi- 
nous, but  shines  by  reflected  sunlight.  If  the  moon  were 
incandescent,  like  the  sun,  we  should  see  it  always  as  a  full- 
moon,  or  complete  luminous  circle.  But  it  is  a  globe,  and 
so  only  one-half  its  surface  can  be  illuminated  by'  the  sun 
at  any  given  moment.  Now  if  the  earth  happens  to  be  so 

162 


Photo  by  Barnard. 
PLATE   6.     Full  Moon  and  Crescent  Moon. 


MOONSHINE 


placed  that  we  can  see  the  entire  illuminated  hemisphere, 
full-moon  occurs.  If  the  earth  is  so  situated  that  we  see 
only  the  unlighted  hemisphere,  the  moon  is  wholly  invisible, 
and  we  say  it  is  " new-moon." 

Evidently,  we  shall  see  the  illuminated  hemisphere  when 
we  are  on  that  side  of  the  moon  which  faces  the  sun  and 
receives  light ;  when  we  are  on  the  side  of  the  moon  opposite 
the  sun,  we  see  the  dark  part.  And  as  the  moon  goes  com- 
pletely around  the  earth  with  respect  to  the  sun  in  29  f 
days,  it  must  happen  once  in  each  such  period  that  we  are 
suitably  placed  for  each  of  these  phenomena.  And,  of 
course,  at  intermediate 
dates,  we  must  be  so 
placed  as  to  see  larger  or 
smaller  portions  of  the 
illuminated  part,  giving 
rise  to  the  other  visible 
phases.  This  is  the 
simple  explanation  first 
found  by  Aristotle. 

It  follows  from  the  above,  as  shown  in  Fig.  40,  that  full- 
moon  must  always  occur  when  the  sun  and  moon  are  seen 
projected  at  nearly  opposite  parts  of  the  celestial  sphere. 
The  figure  shows  how  light  from  the  sun  illumines  half  of 
both  earth  and  moon.  To  the  inhabitants  of  the  dark  side 
of  the  earth,  the  sun  is  not  visible,  and  it  is  night.  But 
those  same  inhabitants  evidently  see  the  bright  half  of  the 
moon,  in  the  full-moon  phase.  Under  these  circumstances, 
the  figure  shows  that  the  directions  of  the  sun  and  full- 
moon,  as  seen  from  the  earth,  point  toward  opposite  sides 
of  the  sky,  approximately. 

It  may  be  remarked,  also,  that  if  the  sun,  earth,  and  moon 

163 


FIG.  40.     Full  Moon. 


ASTRONOMY 

were  always  in  a  single  plane,  the  earth,  at  the  time  of  full- 
moon,  would  be  exactly  in  line  between  the  moon  and  sun. 
It  would  then  cut  off  the  solar  light  from  the  moon  and  give 
rise  to  the  phenomenon  called  an  Eclipse  of  the  Moon. 
But  we  have  already  seen  that  the  two  orbit  planes  are 
not  identical ;  that  there  is  an  angle  of  5°  between  them. 
It  is  this  angle  between  the  planes  that  prevents  the  occur- 
rence of  an  eclipse  during  every  29|-day  period  of  lunar 
orbital  motion,  as  will  be  more  fully  explained  in  a  later 
chapter.  Finally,  Fig.  40  shows  that  the  interval  between 
two  successive  full-moons  or  two  successive  new-moons  is 
the  synodic  period  of  29 1  days,  not  the  sidereal  period  of 
27J  days.  For  these  phases  must  recur  when  the  moon 
has  made  a  complete  revolution  around  the  earth,  measured 
by  the  sun,  not  by  a  star. 

Closely  connected  with  lunar  phases  is  the  phenomenon 
called  the  "earth-shine,"  or  "the  old  moon  in  the  new-moon's 
arms,"  shown  in  Plate  6,  p.  162,  small  photograph.  It 
often  happens  that  when  the  first  slender  lunar  crescent  is 
seen,  a  few  days  after  the  date  of  new-moon,  the  dark  part 
of  the  moon,  within  the  horns  of  the  crescent,  will  be  illu- 
minated faintly.  This  illumination  of  the  dark  part  can- 
not come  directly  from  the  sun,  under  our  accepted  theory 
of  lunar  phases ;  nor  can  it  be  light  from  the  moon  itself, 
for  we  know  the  moon  to  be  non-luminous.  But  it  is  ex- 
plained easily  if  we  once  more  examine  Fig.  40,  p.  163. 
This  figure  makes  clear  that  when  we  see  the  moon  in  the 
full-moon  phase,  the  earth  turns  its  dark  side  toward  the 
moon.  As  seen  from  the  moon,  the  earth  is  in  the  "new- 
earth"  phase. 

All  the  earth  phases,  as  seen  from  the  moon,  are  opposite 
to  the  lunar  phases,  as  seen  from  the  earth.  Thus,  when 

164 


MOONSHINE 

we  see  the  moon  nearly  new,  as  a  slender  crescent,  the  earth 
is  nearly  a  full-earth  to  the  moon.  And  the  slight  illumina- 
tion of  the  dark  part  of  the  moon,  as  we  see  it,  is  then  due 
to  the  strong  light  thrown  upon  it  by  the  brilliant  full-earth, 
doubtless  several  times  more  luminous  than  the  full-moon 
seems  to  us. 

The  small  photograph  of  Plate  6,  p.  162,  also  gives  a 
good  opportunity  to  notice  that  the  " horns"  of  the  moon 
always  appear  to  be  turned  directly  away  from  the  sun,  as 
they  are  seen  by  us  projected  on  the  sky.  This  follows 
from  the  explanation  of  phases :  we  can  understand  it 
easily,  if  we  paint  a  ball  half  black  and  half  white,  to  repre- 
sent the  moon,  with  half  its  surface  illuminated  by  the  sun. 
If  we  now  hold  this  ball  so  as  to  see  only  a  narrow  sickle  of 
the  white  half,  we  shall  always  find  the  horns  of  that  sickle 
turned  to  the  right,  if  the  white  half  of  the  ball,  which 
faces  the  sun,  is  turned  to  the  left. 

Now  the  small  photograph  of  Plate  6  was  made  by  Bar- 
nard, at  the  Yerkes  Observatory  near  Chicago,  Feb.  14, 
1907,  about  one  hour  and  twenty  minutes  after  sunset, 
when  the  moon  was  very  near  the  western  horizon,  where 
the  sun  had  set.  So,  in  Plate  6,  if  we  imagine  a  line  drawn 
between  the  two  ends  of  the  moon's  horns,  and  a  second 
line  perpendicular  to  it,  and  passing  downward  in  the  Plate, 
this  second  line,  if  drawn  far  enough  below  the  horizon, 
would  pass  through  the  sun  on  the  sky. 

The  small  photograph,  therefore,  appears  on  Plate  6  just 
as  it  appeared  to  the  eye  when  Barnard  photographed  it. 
The  large  photograph  was  taken  with  a  different  instrument 
at  a  different  time,  but  it  has  been  purposely  turned  around 
in  Plate  6  to  agree  with  the  small  photograph.  This  agree- 
ment may  be  verified  readily  by  comparing  the  configuration 

165 


ASTRONOMY 

of  markings  on  the  two  pictures.  The  photograph  of  Plate  3, 
p.  17,  shows  the  moon  as  it  would  be  seen  on  the  meridian 
with  an  astronomical  telescope;  to  make  the  large  photo- 
graph of  Plate  6  agree  with  it,  it  would  be  necessary  to  turn 
Plate  6  around  through  more  than  a  right  angle  in  the  direc- 
tion in  which  the  hands  of  a  watch  move.  The  configura- 
tion of  markings  would  then  again  be  in  agreement. 

Having  now  explained  briefly  some  of  the  lunar  phenom- 
ena of  phases  and  motions,  let  us  next  consider  a  pecul- 
iarity in  which  the  moon  differs  absolutely  from  the  earth. 
Astronomers  have  ascertained  quite  definitely  that  lunar 
air  or  atmosphere  is  altogether  absent ;  or,  if  present,  exists 
only  in  an  extremely  attenuated  form.  The  principal  obser- 
vational proof  of  the  non-existence  of  atmos- 
phere is  derived  from  phenomena  known  as 
"occupations"  of  stars.  We  have  seen  that 
FIG.  4i.  Oc-  the  moon,  as  it  moves  in  its  orbit  around  the 

cultations. 

earth,  travels  among  the  stars  about  13  daily 
(p.  161).  But  the  stars  are  very  much  more  distant  than 
the  moon,  though  we  see  both  stars  and  moon  alike  pro- 
jected on  the  background  of  the  celestial  sphere. 

Therefore  it  must  happen  occasionally  that  the  moon 
passes  between  us  and  some  individual  star.  In  such  a 
case  that  star  is,  of  course,  concealed  from  our  view  tem- 
porarily. Usually  such  "  occultations "  last  about  an  hour, 
the  duration  varying  according  to  the  part  of  the  moon  the 
stars  happen  to  meet.  In  Fig.  41,  the  moon  moves  across 
the  sky  in  the  direction  of  the  arrow.  The  star  S  will  there- 
fore be  occulted  longer  than  the  star  Sf,  because  it  meets  a 
wider  part  of  the  disk  of  the  moon.1 

1  The  two  lines  shown  in  the  figure,  along  which  the  two  stars  are  about 
to  be  occulted,  are  called  "chords"  of  the  moon's  disk. 

166 


MOONSHINE 

Now  we  find  from  telescopic  observation  that  no  matter 
where  the  occultation  takes  place,  the  disappearance  of  the 
star  is  always  perfectly  instantaneous ;  there  is  no  gradual 
fading  away ;  it  is  blotted  out  with  very  striking  suddenness 
while  still  in  full  brilliancy.  If  there  were  a  lunar  atmos- 
phere, we  would  surely  see  a  progressive  dimming  of  the 
star,  particularly  as  it  passed  from  the  outer  less  dense 
layers  of  lunar  air  into  the  denser  layers  near  the  surface. 

Knowing,  then,  that  the  moon  has  no  atmosphere,  we 
must  inquire  what  has  become  of  it.  For  we  now  accept 
the  plausible  theory  that  the  moon  was  once  part  of  the 
earth,  and  that  it  was  separated  from  the  parent  planet  as 
a  result  of  a  continued  and  peculiar  action  of  gravitational 
forces.  In  that  case  the  moon  must  have  taken  some 
atmosphere  with  it  when  it  left  the  earth.  What  has  be- 
come of  that  atmosphere? 

The  most  plausible  explanation  of  its  loss  is  derived  from 
the  kinetic  theory  of  gases.  According  to  that  theory, 
the  molecules  of  a  gas  are  in  constant  violent  motion,  and 
continually  colliding  with  each  other.  If  this  was  true  on 
the  outer  confines  of  the  moon's  original  gaseous  atmosphere, 
it  must  have  frequently  happened  that  an  outer  molecule, 
after  collision,  bounced  off  in  a  direction  away  from  the 
moon.  It  then  encountered  no  other  molecule,  and  was 
prevented  from  escaping  into  space  by  nothing  but  the 
moon's  gravitational  attraction.  That  is  the  only  force  to 
hold  it. 

But  the  moon's  gravitational  attraction  is  comparatively 
slight,  as  compared  with  the  earth's ;  for,  as  we  shall  see 
later,  the  mass  of  the  moon  is  only  about  ^\  part  of  the 
earth's  mass;  and  gravitational  attraction  varies  propor- 
tionally to  the  mass  of  the  attracting  body.  Therefore  it 

167 


ASTRONOMY 

is  quite  conceivable  that  the  moon  may  have  lost  its  at- 
mosphere by  the  kinetic  method,  while  the  earth,  by  reason 
of  superior  gravitational  attraction,  is  able  to  retain  it. 
However  this  may  be,  physicists  are  now  agreed  that  there 
is  ample  molecular  velocity  to  carry  gases  gradually  away 
from  the  moon. 

Absence  of  atmosphere  means  also  absence  of  water; 
for  water,  if  present,  would  evaporate  and  form  an  atmos- 
phere. And  without  air  and  water,  there  can  be  no  lunar 
inhabitants  similar  to  ourselves. 

The  shape  of  the  lunar  orbit  around  the  earth,  to  which 
we  have  already  referred,  might  be  ascertained  observa- 
tionally  in  the  manner  already  explained  for  the  earth's 
orbit  around  the  sun  (p.  117).  It  would  merely  be  neces- 
sary to  measure  frequently  the  lunar  angular  diameter,  and 
the  moon's  exact  place  as  projected  on  the  sky  with  refer- 
ence to  the  celestial  equator;  in  other  words,  the  moon's 
declination  and  right-ascension.  This  would  enable  us 
again  to  draw  the  outline  of  an  orbit  similar  geometrically 
to  the  moon's  actual  orbit.  But  as  in  the  case  of  the  earth's 
path,  observations  of  this  kind  give  us  no  notion  as  to  the 
actual  size  of  the  orbit  in  miles.  To  know  this  we  must 
measure  a  linear  distance  somewhere,  just  as  we  found 
when  describing  the  similar  state  of  affairs  in  connection 
with  the  earth's  path  around  the  sun. 

We  shall  therefore  next  outline  the  method  by  which 
this  may  be  done  in  the  case  of  the  moon.  The  easiest 
way  is  to  observe  the  moon's  position,  as  projected  on  the 
sky,  simultaneously  from  two  observatories  widely  sepa- 
rated on  the  earth.  We  can  then  use  the  known  distance 
between  the  two  observatories  as  a  " base-line"  for  calculat- 
ing the  moon's  distance.  Nor  is  it  difficult  to  show  that 

168 


MOONSHINE 

such   calculations   will  make  this  distance  known.     It   is 
found  to  be  jtbout  240,000  miles^ 

It  is  interesting  to  note  that  we  have  now  for  the  first 
time  outlined  a  method  of  finding  by  observation  the  actual 
distance  separating  a  heavenly  body  from  the  earth.  We 
now  see  that  astronomy  can  make  measurements  other 
than  mere  angular  diameters  and  angular  distances.  Its 
grasp  extends  outward  into  space;  by  indirect  methods, 
but  methods  perfectly  valid,  man  has  learned  the  distance 
of  the  moon  just  as  though  he  could  go  there  and  measure  it 
with  a  surveyor's  tape-line. 

Closely  related  to  the  method  of  ascertaining  the  moon's 
distance  is  the  mysterious  word  "  parallax."  The  moon's 
parallax  is  defined  as  the 
angular  semi-diameter  or 
radius  of  the  earth,  as  seen 
from  the  moon.  Thus,  in 

_.  FIG.  42.    Parallax  of  the  Moon. 

Fig.    42,    AC   is   the  earth's 

radius ;   M  is  the  moon ;   and  the  small  angle  at  M  is  the 

lunar  parallax.2 

The  moon's  distance  (240,000  miles,  in  round  numbers) 
is  about  60  times  the  earth's  radius.  But  of  course  the 
flattening  of  the  lunar  orbit  makes  the  distance  vary,  just 
as  we  found  was  the  case  with  the  earth  and  sun,  when  we 
discussed  the  terrestrial  seasons.  Just  as  the  earth  has  a 
perihelion,  or  nearest  approach  to  the  sun  (p.  120),  so  the 
moon  has  a  " perigee,"  or  nearest  approach  to  the  earth. 
And  the  lunar  orbit  is  more  flattened  than  that  of  the 
earth ;  the  actual  distance  of  the  moon  may  vary  all  the 
way  from  222,000  to  253,000  miles. 

The  axial  rotation  of  the  moon  is  a  subject  often  found 

1  Note  19,  Appendix.  *2  Note  20,  Appendix. 

169 


ASTRONOMY 

puzzling,  though  really  very  simple.  Here  the  crucial  fact  is 
derived  from  the  most  elementary  telescopic  observation 
of  the  moon.  We  find  that  the  moon  always  turns  approxi- 
mately the  same  hemisphere  toward  the  earth.  Whenever 
we  look  at  the  moon,  we  see  the  same  configuration  of  surface 
details,  lunar  mountain  ranges,  etc. ;  we  never  see  the  moun- 
tain ranges  on  the  moon's  opposite  side. 

There  can  be  but  one  reasonable  explanation  of  this. 
The  moon  must  have  an  axial  rotation  just  rapid  enough  to 
produce  this  peculiar  result.  And  here  is  the  puzzle : 
many  persons  ask  how  the  moon  can  have  any  axial  rotation 
at  all,  if  it  constantly  turns  the  same  face  toward  us.  The 
matter  will  be  understood  most  easily  by  means  of  a  simple 
experiment.  Let  the  reader  face  a  table  in  the  middle  of  a 
room.  Let  him  imagine  himself  to  be  the  moon,  and  the 
table  to  be  the  earth.  Let  him  now  walk  around  the  table 
in  such  a  way  that  he  faces  it  constantly.  When  he  has 
gone  halfway  -around  the  table,  always  facing  it,  he  will  find 
that  he  is  looking  at  that  wall  of  the  room  toward  which  his 
back  was  turned  when  he  began  the  experiment. 

Thus  he  must  have  turned  himself  halfway  around,  while 
constantly  facing  the  table.  If  his  face  was  turned  toward 
the  north  when  he  began,  it  is  now  turned  toward  the  south. 
And  if  he  completes  a  circuit  of  the  table  in  the  same  way, 
returning  finally  to  his  original  position,  he  will  find  that  he 
has  faced  successively  every  point  of  the  compass.  This 
proves  that  he  has  turned  himself  around,  or  rotated  once  on 
his  vertical  axis ;  yet,  representing  the  moon,  he  has  at  all 
times  turned  his  face  toward  the  table,  representing  the 
earth. 

Accurately  stated,  the  case  stands  thus  :  the  moon  makes 
an  axial  rotation  in  exactly  the  same  time  it  takes  to  make  an 

170 


MOONSHINE 

orbital  revolution  around  the  earth.  We  have  seen  that 
it  revolves  in  its  orbit  in  27^  days ;  it  also  finishes  a  rota- 
tion on  its  axis  in  27^  days.  This  is  the  whole  explanation 
of  the  mystery. 

To  complete  this  matter  of  the  moon's  rotation,  we  must 
now  point  out  that  the  explanation,  so  far  given,  is  not  quite 
exact,  though  it  is  very  nearly  so,  and  quite  sufficiently  so 
for  a  first  approximation.  There  is  a  phenomenon  called 
Libration  of  the  moon,  which  makes  it  possible  for  us  to 
see  a  somewhat  different  part  of  the  lunar  surface  at  certain 
times.  The  lunar  rotation  axis  is  slightly  inclined  from 
perpendicularity  to  the  plane  in  which  is  situated  the  orbit 
of  the  moon  around  the  earth.  The  inclination  is  small, 
about  6^° ;  but  it  has  the  effect  of  tilting  the  moon,  as  it  were, 
6J°,  first  one  way  and  then  the  opposite  way,  according  to  its 
position  in  its  orbit  around  the  earth.  For  the  lunar  rota- 
tion axis  remains  constantly  parallel  to  its  original  direction 
in  space  during  the  entire  orbital  revolution.  On  account 
of  this  tilting  effect,  we  see  a  slightly  different  hemisphere 
of  the  moon  at  different  dates  in  each  lunar  period. 

Still  another  libration  of  the  moon  exists.  It  is  true  that 
the  moon  rotates  on  its  axis  in  the  same  period  of  time  it 
requires  for  its  orbital  revolution  around  the  earth;  but 
while  the  axial  rotation  is  uniform,  the  orbital  motion, 
of  course,  is  variable.  As  in  all  orbital  motion,  the  velocity 
is  greatest  when  the  moon  is  in  that  part  of  its  orbit  which 
lies  nearest  the  earth.  Consequently,  the  axial  rotation 
and  the  orbital  revolution  do  not  increase  at  precisely  the 
same  rate ;  and  from  this  cause,  also,  we  see  a  slightly  different 
hemisphere  of  the  moon  at  different  dates  in  the  month. 

These  are  the  two  principal  librations;  there  remain 
certain  other  very  slight  ones  which  we  may  here  omit 

171 


ASTRONOMY 

as  unimportant.     But  the  combined  effect  of  them  all  is 
as  follows : 

of  the  lunar  surface  is  always  visible, 

of  the  lunar  surface  is  never  seen, 

°f  the  lunar  surface  is  sometimes  visible. 

It  is  of  interest  to  note  in  passing  that  this  agreement  of 
the  lunar  axial  rotation  period  with  the  period  of  orbital 
revolution  is  not  due  to  chance.  It  must  have  resulted  from 
some  physical  cause;  the  theory  at  present  accepted  by 
astronomers  considers  it  to  be  a  result  of  forces,  inter- 
acting between  the  moon  and  the  earth,  and  analogous 
to  those  producing  ocean  tides.  These  forces  probably 
brought  about  the  existing  state  of  things  long  ago,  in 
early  cosmic  ages,  when  the  moon  may  be  considered  to 
have  not  yet  become  perfectly  solid,  and  to  have  therefore 
been  subject  to  enormous  tidal  distortions. 

The  next  thing  we  have  to  do  is  to  explain  how  astronomers 
measure  and  weigh  the  moon.  Of  course  this  cannot  be 
done  by  the  methods  used  in  the  case  of  the  earth,  because 
it  is  impossible'  to  visit  the  moon  to  make  surveys  and 
perform  the  Cavendish  experiment  (p.  107)  for  determining 
mass.  But  the  moon's  size  can  be  derived  easily  from 
measures  of  its  angular  diameter,  combined  with  the  knowl- 
edge we  have  already  obtained  as  to  its  distance  from  the 
earth. 

In  Fig.  43,  the  angle  AEB  is  the  moon's  angular  diameter 
(p.  118),  as  seen  from  thq  earth  E.  BE  and  AE  are  each 
equal  to  the  known  distance  of  the  moon  from  the  earth.  The 
triangle  ABE  is  therefore  fully  known,1  and  we  can  calculate 

1  All  parts  of  a  triangle  can  be  calculated  by  trigonometric  methods,  if 
we  know  two  sides  and  the  angle  between  them. 

172 


MOONSHINE 

the  number  of  miles  in  the  lunar  diameter  AB.  The 
average  angular  diameter  is  measured  easily  with  astronomi- 
cal instruments;  it  is  found  to  be  about  31 '  of  arc.  This, 
combined  with  the  known  distance  (about  240,000  miles), 
makes  the  moon's  diameter  about  2200  miles,  or  a  little  more 
than  one-quarter  of  the  earth's  diameter.  And  as  we  know 
from  geometry  that  the  volumes  of  spheres  are  proportional 
to  the  cubes  of  their 
diameters,  it  follows 
that  the  volume  of 
the  earth  is  some- 

FIG.  43.    Moon's  Diameter. 

what  more  than  64 

times  that  of  the  moon  (64  =  4  X  4  X  4).     More  accurately 

stated,  the  earth's  volume  is  about  50  times  that  of  the 

moon. 

A  somewhat  more  difficult  problem  is  the  " weighing"  of 
the  moon,  which,  as  we  have  already  seen  in  the  case  of 
the  earth  (p.  103),  really  means  a  determination  of  the 
moon's  mass.  Curiously  enough,  the  mass  of  the  moon  is 
most  simply  determined  by  observations  of  the  sun.  To 
understand  how  this  is  done,  we  must  begin  by  correcting 
an  approximately  accurate  theory  which  we  have  so  far 
found  sufficient  for  our  explanations.  It  is  frequently 
convenient,  for  the  sake  of  lucidity,  to  begin  the  explanation 
of  some  phenomenon  by  assuming  a  state  of  affairs  resembling 
closely  that  which  actually  exists  in  nature,  and  afterwards 
substituting  new  explanations  successively,  each  more 
closely  approximating  to  the  truth,  until  we  can  finally  con- 
sider a  tolerably  complete  theory  in  its  full  complexity. 

In  the  present  instance,  we  must  now  correct  a  previous 
statement  concerning  the  earth's  orbit  around  the  sun.  The 
earth  has  so  far  been  said  to  pursue  an  oval  or  elliptic  orbit, 

173 


ASTRONOMY 

with  the  sun  at  one  focus.  And  by  the  earth  we  mean, 
of  course,  the  earth's  center.  But  we  now  know  that 
the  earth  and  moon  together  are  traveling  in  that  orbit 
around  the  sun.  Therefore,  speaking  accurately,  it  is  not 
the  earth's  center  that  is  exactly  in  the  orbit,  but  rather 
the  combined  " center  of  gravity"  of  the  earth  and  moon. 
And  by  center  of  gravity  we  mean  a  point  so  situated  on  the 
line  joining  earth  and  moon  that  their  weights,  as  it  were, 
would  just  balance  about  the  center  of  gravity.  Figure  44 
shows  this  position  of  the  center  of  gravity  at  c.  If  we  imag- 
ine the  earth  and  moon  attached  to  the  ends  of  a  rigid  bar 
240,000  miles  long,  their  weights  would  balance  if  the  bar 

were  supported  at  the  point  c. 
r\   And  owing  to  the  great  mass 


Moon    of  the  earth  as  compared  with 

Center  of  Gravity  of  Ear.n        the      mOOn,      this      Center      of 

and  Moon. 

gravity  is  much  nearer  the 

earth's  center  than  the  moon's.     It  is,  in  fact,  inside  the 
earth's  surface. 

Now  this  center  of  gravity  has  another  peculiarity  of  the 
utmost  importance.  Not  only  is  it  the  point  that  is  really 
following  out  the  terrestrial  orbit  around  the  sun,  but  it  is 
also  the  real  focus  about  which  the  moon  pursues  its  monthly 
orbit  around  the  earth.  The  moon,  accurately  speaking, 
does  not  revolve  about  the  earth,  but  about  the  point  c,  the 
center  of  gravity  of  the  earth  and  moon  combined.  Further- 
more, while  the  moon  is  going  around  this  center,  the 
earth  is  doing  the  same  thing,  though  in  a  much  smaller 
orbit.  Again  imagining  both  bodies  attached  to  the  ends  of 
a  rigid  rod,  it  is  a  little  as  though  this  rod  were  pivoted  at 
the  center  of  gravity,  and  turning  around  it.  Thus  the 
force  of  gravitation  causes  both  bodies  to  revolve 

174 


MOONSHINE 

about  their  common  center  of  gravity,  but  little  moon  can- 
not make  big  earth  travel  in  as  large  an  orbit  as  big  earth 
imposes  on  little  moon. 

The  final  result  is  to  swing  the  earth  each  half  month  a 
short  distance  either  forward  or  backward  with  respect  to 
the  position  it  would  occupy  in  its  annual  orbit  around  the 
sun,  if  there  were  no  moon.  Sometimes  the  earth  is  in 
advance  of  the  center  of  gravity;  sometimes  behind  it. 
But  we  always  see  the  sun  projected  on  the  celestial  sphere 
at  a  point  on  the  ecliptic  circle  directly  opposite  the 
earth's  actual  position  in  its  orbit ;  therefore  this  center 
of  gravity  effect  must  show  itself  by  slightly  advancing 
or  retarding  the  sun's  apparent  motion  in  the  ecliptic 
circle. 

The  whole  phenomenon  is  very  slight,  amounting  to  a 
total  change  in  the  sun's  apparent  place  on  the  ecliptic 
circle  of  only  12"  of  arc.  Yet  this  can  be  measured  with 
accurate  instruments ;  and  a  simple  calculation  then  shows 
that  the  common  center  of  gravity  of  earth  and  moon  is 
distant  only  2880  miles  from  the  earth's  center.  This  is 
about  gV  Part  of  the  total  distance  between  the  centers  of 
these  two  bodies ;  therefore  the  lunar  mass  must  be  about 
gy  part  of  the  earth's  mass.1 

Having  thus  found  the  moon's  volume  to  be  about  -^  that 
of  the  earth,  and  its  mass  only  ^T,  it  follows  that  the  moon 
must  on  the  average  be  composed  of  materials  less  dense 
than  those  of  which  the  earth  is  made.  If  the  moon  were 
equally  dense  with  the  earth,  a  cubic  foot  of  average  lunar 
material  would  weigh  as  much  as  a  cubic  foot  of  average 
terrestrial  material ;  and  these  ratios  of  masses  and  volumes 
between  the  two  bodies  would  be  equal.  The  figures  we  have 

1  Note  21,  Appendix.   x 

175  '  **< 


ASTRONOMY 

obtained  make  the  moon's  density  about  six-tenths  that 
of  the  earth. 

The  interval  of  time  between  two  successive  returns  of  the 
moon  to  the  meridian  of  any  place  on  the  earth  may  be  called 
the  Lunar  Day.  Its  length  depends  on  the  diurnal  axial 
rotation  of  the  earth,  in  a  manner  analogous  to  the  relation 
between  sidereal  and  solar  days  (p.  65).  We  have  seen 
that  the  sidereal  day  is  equal  to  the  period  of  the  earth's 
axial  rotation,  and  is  therefore  the  interval  of  time  between 
two  successive  returns  of  the  vernal  equinox  to  the  me- 
ridian. We  have  also  seen  that  the  solar  day  is  about  four 
minutes  longer  than  the  sidereal  day,  because  the  sun's 
apparent  daily  motion  of  one  degree  along  the  ecliptic 
circle  makes  the  sun  lag  a  little  behind  the  equinox  point, 
so  that  the  apparent  rotation  of  the  heavens  must  continue 
about  four  minutes  after  each  complete  axial  rotation  of  the 
earth,  to  enable  the  sun  to  reach  the  meridian  again  (p.  69). 
The  case  of  the  moon  is  precisely  similar ;  only,  as  its  daily 
motion  averages  about  13°  instead  of  1°,  the  excess  length  of 
the  lunar  day  is  about  52  minutes,  instead  of  4  minutes. 
This  makes  the  lunar  day  average  about  24h  52m  of  sidereal 
time. 

The  fact  that  the  moon  thus  reaches  the  meridian  about 
52  minutes  later  each  night  means  that  it  will  also  rise  and 
set  about  52  minutes  later  each  night.  But  this  is  only  an 
average  figure;  in  the  latitude  of  New  York,  for  instance, 
the  daily  retardation  of  moonrise  may  vary  all  the  way  from 
23  minutes,  to  1  hour  17  minutes. 

When  this  retardation  of  the  time  of  moonrise  is  at  the 
minimum  of  23  minutes,  the  moon  will  rise  at  nearly  the 
same  time  on  two  or  three  successive  nights.  If  the 
moon  also  happens  to  be  almost  a  full-moon  on  such  an 

176 


MOONSHINE 

occasion,  we  have  the  phenomenon  known  as  the  Harvest 
Moon.  This  is  defined,  then,  as  the  rising  of  the  moon, 
nearly  full,  on  two  or  three  successive  nights  at  nearly  the 
same  hour. 

To  ascertain  when  this  will  occur,  we  must  discuss  the 
principal  cause  of  these  large  variations  in  the  daily  retarda- 
tion of  the  time  of  moonrise.  For  this  purpose  we  may,  with 
sufficiently  close  approximation,  consider  the  moon  as 
appearing  always  in  the  ecliptic  circle  on  the  sky;  as  we 
already  know,  it  is  actually  never  very  far  from  that  circle. 
This  being  premised,  it  is  clear  that  the  time-interval  be- 
tween the  moonrises  on  two  successive  nights  will  depend 
on  the  angle  between  the  ecliptic 
circle  and  the  horizon,  as  shown 
in  Fig.  45.  HH  is  part  of  the 
horizon ;  VV  part  of  the  ecliptic 
circle.  Let  us  suppose  the  moon 

rlG.  45. 

was  at  the  intersection  /  when 
it  rose  on  a  certain  night.  Exactly  twenty-four  hours  later 
the  point  /  will  be  again  rising  above  the  horizon  HH.  But 
in  those  twenty-four  hours  the  moon  will  have  moved  along 
the  ecliptic  to  the  point  /',  about  13°  from  /. 

How  much  later  will  the  moon  rise  on  the  second  night  ? 
Clearly,  by  a  time-interval  exactly  equal  to  the  time  in  which 
the  apparent  rotation  of  the  celestial  sphere  will  move 
the  point  /'  up  to  the  horizon  HH.  This  interval  will  be 
short,  if  the  angle  HIV  between  the  horizon  and  ecliptic  is 
small. 

But  the  angle  HIV  is  not  always  the  same.     It  is  easy  to 

demonstrate,  by  the  aid  of  a  celestial  globe,  that  it  is  a 

minimum  when  the  point  of  intersection  7  is  at  the  vernal 

equinox  (p.  35).     This  is  well  illustrated  by  the  small  photo- 

N  177 


ASTRONOMY 

graph  of  Plate  6,  p.  162.  The  line  joining  the  moon's  horns 
being  nearly  horizontal,  the  ecliptic  must  be  nearly  perpen- 
dicular to  the  horizon  if  the  horns  are  to  point  directly  away 
from  the  sun  (p.  165).  And  the  date  of  the  photograph 
being  near  the  vernal  equinox,  about  the  time  of  sunset, 
it  follows  that  the  angle  HIV,  being  nearly  a  right  angle,  is  at 
a  maximum  at  the  western  horizon  on  or  about  March  21. 
Moreover,  near  the  eastern  horizon,  it  will  be  at  a  mini- 
mum on  the  same  date.  That  the  ecliptic  rises  very  high 
from  the  western  horizon  at  sunset  on  March  21  is  also  shown 
by  the  table  on  p.  49. 

It  results  from  these  considerations  that  if  the  full-moon 
occurs  when  the  moon  appears  near  the  vernal  equinox 
point,  the  daily  retardation  of  moonrise  will  be  a  minimum. 
But  we  have  already  found  (p.  163)  that  the  full-moon  always 
necessarily  appears  opposite  the  sun  in  the  sky.  Therefore, 
on  the  occasion  of  a  harvest  moon,  the  sun  must  be  at  the 
autumnal  equinox  (p.  35),  which  is  directly  opposite  the 
moon's  position  at  the  vernal  equinox.  But  the  sun  appears 
in  the  autumnal  equinox  about  September  22  in  each  year. 
Consequently,  the  harvest  moon  is  always  the  full-moon 
which  happens  nearest  to  September  22. 

And  this  explains  the  name  " harvest."  For  certain 
harvests  are  gathered  in  September ;  and  it  is  of  consequence 
to  farmers  to  have  plenty  of  moonlight,  so  that  their  work 
may  be  completed  before  rain  falls.  The  full-moon,  being 
opposite  the  sun,  will  rise  when  the  sun  sets,  which  occurs 
at  six  o'clock  on  the  day  of  the  equinox.  Thus  the  harvest 
full-moon  will  rise  on  two  or  three  consecutive  dates  at  about 
six  in  the  evening,  and  will  remain  visible  until  sunrise  the 
next  morning. 

Still  another  phenomenon  of  interest  arises  from  the  fact 

178 


MOONSHINE 

that  the  full-moon  always  appears  opposite  the  sun  in  the 
sky.  Near  the  time  of  the  winter  solstice  (p.  121)  in  Decem- 
ber the  full-moon  must  be  near  the  summer  solstice  point 
of  the  ecliptic  circle,  in  order  that  it  may  be  opposite  the 
sun.  It  follows  from  this  that  the  winter  full-moons  appear 
far  north  of  the  celestial  equator,  like  the  sun  in  summer. 
Consequently,  the  full-moon  in  winter  "rides  high/'  as  the 
saying  is;  when  on  the  meridian  it  will  appear  near  the 
zenith,  while  the  summer  full-moons  are  low  down  in  the 
sky,  like  the  sun  in  winter. 

These  variations  in  the  time  of  moonrise  are  always  set 
forth  in  ordinary  almanacs ;  but  a  certain  peculiarity  of  this 
part  of  the  almanacs  requires  explanation.  In  the  case 
of  the  sun,  the  almanacs  give  both  the  time  of  sunrise  and  sun- 
set, all  of  which  is  understood  without  difficulty.  But  for 
the  moon,  the  almanacs  give  only  the  time  of  rising  or  the 
time  of  setting,  —  never  both.  And  both  are  not  needed. 
If  the  moon,  for  instance,  rises  shortly  after  sunset, 
it  will  set  shortly  after  the  next  sunrise.  It  will  therefore 
be  in  the  sky  »when  the  sun  rises,  and  will  set  during  day- 
light, —  a  phenomenon  not  usually  observable.  In  other 
words,  only  one  of  the  two  phenomena,  moonrise  or  moonset, 
can  be  observed  on  any  given  date,  and  the  almanac  always 
gives  the  time  of  the  observable  phenomenon. 

But  this  introduces  another  complication.  As  the  lunar 
"day"  is  24h  52m  long,  it  may  happen  now  and  then  that  a 
given  solar  day  of  24  hours  contains  no  moonrise  at  all. 
The  moon  might  have  risen  just  before  the  beginning  of  the 
solar  day,  and  might  rise  again  just  after  the  ending  of  it. 
In  fact,  this  must  occur  once  each  month.  If  the  moon- 
column  *in  the  almanac  contains  the  word  "rises,"  the  follow- 
ing numbers  in  the  column  are  the  successive  times  of  moon- 

179 


ASTRONOMY 

rise.  On  the  date  when  the  moon  does  not  rise,  the  abbre- 
viated word  "morn"  is  then  substituted  in  the  moon  column 
for  the  usual  time  of  moonrise.  The  following  numbers  in 
the  column  then  indicate  that  the  moon  rises  after  midnight, 
—  in  the  morning. 

In  the  lunar  orbit  there  exists  still  one  more  peculiarity 
that  illustrates  the  tendency  of  astronomy  to  deceive  us  by 
entangling  the  seeming  and  the  true,  —  a  tendency  that 
has  much  to  do  with  the  peculiar  fascination  of  the  science. 
To  an  observer  on  the  earth  the  moon's  orbit  seems  to  be 
an  ellipse  or  oval  curve ;  but  the  true  orbit  is  not  really  an 
ellipse  at  all.  For  while  the  moon  is  traveling  around  the 
earth,  the  earth  is  itself  speeding  through  space  in  its  annual 
orbit  around  the  sun,  dragging  with  it  the  moon  and  the 
lunar  orbit  around  the  earth. 

Consequently,  though  the  moon's  orbit  is  an  ellipse,  so 
far  as  we  dwellers  on  the  earth  are  concerned,  its  real  path  in 
space  is  compounded  of  the  two  motions  involved :  first, 
the  lunar  motion  around  the  earth ;  and  second,  the  terrestrial 
motion  around  the  sun.  Now  the  earth's  linear  velocity  of 
motion  around  the  sun  is  much  more  rapid  than  the  lunar 
motion  around  the  earth,  and  is  therefore  of  greater  influence 
in  fixing  the  true  shape  of  the  orbit  in  space.  And  it  is 
known  that  the  sun  is  itself  also  moving  through  space, 
carrying  with  it  the  earth  and  the  whole  solar  system,  in- 
cluding the  moon.  This  motion  would  also  affect  the  shape 
of  the  lunar  orbit ;  but  we  shall  here  consider  only  the  two 
principal  causes  already  mentioned,  —  the  moon's  motion 
around  the  earth,  and  the  earth's  motion  around  the  sun. 

It  is  a  very  singular  thing,  and  one  not  altogether  easy 
to  understand,  that  the  combination  of  these  motions  makes 
the  true  path  of  the  moon  always  concave  toward  the  sun, 

180 


MOONSHINE 

as  shown  in  Fig.  46.  The  arrow  indicates  the  direction  of 
the  sun ;  E\,  E2,  Es,  etc.,  are  five  successive  positions  of  the 
earth  in  its  annual  orbit  around  the  sun,  separated  by  an 
interval  of  about  7^  days,  or  one-quarter  of  a  lunar  synodic 
period  (p.  161).  The  points  Mit  M2,  7lf3,  etc.,  are  five  cor- 
responding positions  of  the  moon.  MI  and  M5  are  new- 
moon  positions ;  Ms  a  full-moon  position ;  M2  and  M4  rep- 
resent quartered  phases.  The  whole  line  MiM2M3M^M^ 
represents  a  part  of  the  moon's  actual  orbit  in  space  with 
respect  to  the  sun ;  and  we  can  prove  without  difficulty 
that  it  is  everywhere  concave  toward  the  sun.1 


FIG.  46.    Moon's  Path  with  Respect  to  the  Sun. 

When  considering  the  lunar  atmosphere  we  found  the 
moon  quite  unlike  the  earth.  But  there  exists  also  a  very 
conspicuous  similarity  between  the  two  bodies,  —  the  moun- 
tainous character  of  their  surfaces.  There  are  a  number 
of  mountain  ranges  on  the  moon,  and  numerous  craters 
apparently  of  volcanic  origin ;  but  there  are  no  active 
volcanoes.  These  lunar  mountains  are  from  1000  to 
2000  feet  high,  and  some  of  the  craters  are  50  miles 
in  diameter.  In  the  center  of  the  crater  there  is 
often  a  conical  mountain  peak;  it  is  as  though  the  crater 
wall  was  formed  by  a  shower  of  volcanic  material  ejected  from 
a  center,  and  falling  in  a  circle  around  it.  The  central 
peak  may  then  have  resulted  from  a  final  outburst  of  the 
volcanic  discharge,  after  the  explosive  force  of  the  volcano 

1  Note  22,  Appendix. 
181 


ASTRONOMY 

had  become  too  feeble  to  throw  its  lava  far  from  the  eruptive 
center.  The  moon's  surface  also  shows  many  "rills"  or 
crooked  valleys  radiating  from  certain  craters.  These 
surface  features  are  well  seen  in  Plate  7.  At  the  bottom  of 
this  photograph  is  the  great  crater  Theophilus,  with  its 
rugged  central  mountain  peak. 

The  height  of  lunar  mountains  and  crater  walls  may  be 
measured  with  the  telescope.  In  certain  lunar  phases,  when 
sunlight  falls  obliquely  on  the  moon's  surface,  the  moun- 
tains cast  long  black  shadows,  seen  conspicuously  in  Plate  7. 
It  is  possible  to  measure  in  the  telescope  the  angular  length 
of  such  shadows ;  and  knowing  the  moon's  distance,  we  can, 
then  calculate  the  shadow  lengths  in  miles  from  the  measured 
angular  lengths.  (Cf.  p.  172.)  Then,  from  the  calculable 
angle  at  which  sunlight  falls  on  the  lunar  surface  at  the 
moment  when  the  shadows  were  measured  in  the  telescope, 
and  the  known  shadow  lengths  in  miles,  we  can  compute 
the  mountain  heights,  also  in  miles,  by  methods  well  known 
to  surveyors. 


182 


PLATE   7.     Lunar  Enlargement. 


Photo  by  Ritchey. 


CHAPTER  XI 

THE    PLANETS 

IN  discussing  the  celestial  sphere  (p.  23)  and  the  ap- 
pearance of  the  stars  projected  upon  it,  we  found  that  the 
great  mass  of  these  luminous  points  retain  practically 
unchanging  relative  positions  on  the  sphere,  and  are  subject 
only  to  such  apparent  motions  as  result  from  the  earth's 
daily  rotation  on  its  axis  and  annual  orbital  revolution 
around  the  sun.  At  the  same  time,  a  certain  small  number 
of  stars  move  about  among  their  fellows  (p.  10).  These 
are  the  "  wanderers/'  —  the  Planets.  Five  are  easily  visible 
to  the  unaided  eye,  —  Mercury,  Venus,  Mars,  Jupiter,  and 
Saturn.  Uranus  may  also  be  seen  without,  a  telescope  under 
favorable  conditions ;  Neptune,  and  the  great  body  of  tiny 
telescopic  objects  of  the  planetary  class,  called  Planetoids, 
require  optical  help  to  be  seen. 

The  distinguishing  thing  about  these  planets  is  that 
they  all  belong  to  our  solar  system.  The  earth  is  merely 
one  of  the  planets  in  that  system ;  the  others,  like  the  earth, 
revolve  around  the  sun  in  orbits  analogous  to  the  earth's 
own  annual  orbit.  These  planetary  orbits  are  all  oval 
or  elliptic,  and  have  the  sun  at  a  point  near  the  center  of 
the  orbit, — the  focus  (p.  116). 

When  explaining  the  earth's  annual  orbital  revolution 
around  the  sun,  we  described  a  simple  method  of  observation 
by  which  the  form  of  the  terrestrial  orbit  might  be  deter- 
mined experimentally.  These  simple  observations  were  also 

183 


ASTRONOMY 


FIG.  47.    Law  of  Areas. 


found  capable  of  establishing  for  the  earth  a  law  of  planetary 
orbital  motion  first  discovered  by  Kepler ;  viz.  that  the 
" radius  vector"  (p.  119),  or  line  joining  the  planet  and  the 

sun,  moves  over  equal  areas  in 
equal  times.  Thus,  in  Fig.  47,  S 
represents  the  sun,  PI,  P2,  P3,  P4, 
four  positions  of  a  planet  in  its 
orbit,  such  that  the  motion  from 
PI  to  P2  is  accomplished  in  the 
same  interval  of  time  required  for 
motion  from  P3  to  P4.  Then  the 
triangular  area  SPiP2,  included  be- 
tween the  two  radii  vectores  SPi, 

£P2,  and  the  arc  of  the  curved  orbit  PiP2,  is  equal  to  the 
other  triangular  area  SPSP^  similarly  included  between  two 
radii  vectores  and  an  arc  of  the  curved  orbit. 

We  must  now  prove  that  this  law  applies  universally  to  all 
planets,  and  that  it  is  a  necessary  consequence  of  Newton's 
law  of  gravitation.  This  latter  law,  as  we  have  already 
seen  (p.  103),  declares  that  an  attraction 
exists  between  the  sun  and  planet,  directly 
proportional  to  the  product  of  their  masses, 
and  inversely  proportional  to  the  square  of 
the  distance  between  thejn. 

Let  us  consider  Fig.  48,  and  suppose 
that  at  a  certain  instant  of  time  the  sun 
is  situated  at  S,  with  the  planet  at  PI  ;  P,« 
and  let  us  first  examine  what  the  planet's 
motion  would  be  if  there  were  no  such 
thing  as  an  attraction  toward  the  sun.  We  may  suppose 
the  planet  to  be  traveling  with  a  certain  velocity,  and  in  a 
certain  direction,  such  as  would  carry  it  to  P2  at  the  end  of 

184 


FIG.  48.    Planetary 
Motion. 


THE   PLANETS 

one  second  of  time.  This  original  velocity  and  motion  may 
be  regarded,  if  we  choose,  as  a  result  of  the  original  cata- 
clysm whereby  the  planet  was  first  brought  into  separate 
existence. 

Now  if  there  were  no  attraction  toward  the  sun,  and  as  there 
can  be  no  friction  or  resistance  to  motion  in  empty  space, 
the  planet  will  arrive  at  P2  endowed  with  the  same  velocity 
and  direction  of  motion  which  it  originally  possessed  at  PI. 
Therefore  it  will,  under  these  circumstances,  travel  an  equal 
distance  along  the  same  straight  line  in  the  next  second, 
and  thus  arrive  at  P3.  The  line  PiP2P3  is  the  planet's 
orbit,  if  there  be  no  attraction  toward  the  sun,  and  the 
lines  SPi,  $P2,  £P3,  are  three  positions  of  the  planet's  radius 
vector. 

The  area  traveled  over  by  the  radius  vector  in  the  first 
second  is  the  triangle  $PiP2 ;  and  in  the  second  second  it 
is  the  triangle  *SP2P3.  But  these  two  triangles  have  equal 
areas;1  and  this  constitutes  a  proof  that  the  radius  vector 
moves  over  equal  areas  in  equal  times,  if  there  exists  no 
attraction  whatever  toward  the  sun. 

Next  suppose  that  the  solar  attraction  exists,  but  that 
instead  of  being  continuous  in  action  it  is  applied  suddenly 
in  the  form  of  an  impulse  toward  the  sun  at  the  end  of  each 
second  of  time.  Suppose  the  first  impulse  is  applied  at  the 
end  of  the  first  second  of  time,  when  the  planet  has  reached 
P2,  and  that  it  is  applied  toward  the  sun  along  the  radius 
vector  P2S.  Now  consider  Fig.  49,  and  imagine  the  im- 
pulse toward  the  sun  strong  enough  to  have  carried  the 
planet  to  P2'  in  one  second  of  time,  supposing  the  said  im- 

1  Readers  familiar  with  geometry  will  recognize  that  these  triangles 
are  equal  because  they  have  a  common  vertex  at  S,  and  equal  bases 
PiP2  and  P2P3  situated  upon  a  single  straight  line. 

185 


ASTRONOMY 

pulse  toward  the  sun  to  have  acted  alone  during  one  second. 
But  the  planet  at  P2  is  also  subject  to  the  original  force, 
which,  acting  alone,  would  have  moved  it  to  P3  in  the  second 
second  of  time.  Thus  the  planet  at  P2  is  subject  to  two 
forces,  one  of  which,  acting  alone,  would  have  carried  it  to 
P2'  in  the  next  second ;  and  the  other,  likewise  acting  alone, 
would  have  carried  it  to  P3  in  that  next  second. 

Where  will  the  planet  go  under  the  combined  action  of 
these  two  forces  in  the  second  second  of  time?  It  must 
evidently  move  along  a  line  intermediate 
in  direction  between  P2P3  and  P2P2'. 
That  line  will  in  fact  be  P2P3/,  and  at 
the  end  of  the  second  second  the  planet 
will  arrive  at  the  point  P/.1  Its  radius 
vector  will  then  be  the  line  $P3'  ;  and 
the  areas  traversed  by  the  radius  vector 
in  the  two  consecutive  seconds  of  time 
here  under  consideration  will  be  the  tri- 
angles SPiP2  and  $P2P3'.  It  is  not  difficult  to  show  that 
the  areas  of  these  two  triangles  are  also  equal.2  Conse- 
quently, under  our  present  supposition  as  to  the  nature  of 
the  attraction  toward  the  sun,  the  planetary  orbit  PiP2P/ 
still  satisfies  the  law  of  areas. 

It  is  evident  that  any  number  of  impulses  toward  the  sun 
at  the  ends  of  other  successive  seconds  of  time  would  pro- 
duce similar  results.  And  the  same  reasoning  would  hold 
true  if  we  suppose  the  impulses  to  occur  more  frequently; 
say  ten  or  a  hundred  times  in  a  second  of  time.  It  follows 
that  if  we  increase  sufficiently  the  number  of  supposed  im- 
pulses per  second,  we  can  at  last  transform  our  orbit  from  a 
series  of  very  short  straight  lines  into  an  actual  curve ;  for 

1  Note  23,  Appendix.  2  Note  24,  Appendix. 

186 


THE  PLANETS 

every  curve  may  be  regarded  as  made  up  of  an  infinite 
number  of  excessively  short  straight  line  elements.  And 
at  the  same  time,  the  supposed  series  of  impulses  toward 
the  sun,  coming  infinitely  close  together,  are  transformed 
into  the  continuous  action  of  gravitational  attraction. 
The  above  reasoning  therefore  constitutes  a  proof  that  a 
planet  moving  under  the  influence  of  an  original  impulse 
in  any  direction,  plus  a  gravitational  attraction  toward 
the  sun,  will  pursue  an  orbit  satisfying  the  law  of  equal 
areas  for  the  radius  vector. 

One  of  the  most  interesting  things  in  the  above  proof  is 
the  absence  of  any  special  requirements  as  to  the  nature  of 
solar  gravitational  attraction.  Nothing  in  the  proof  de- 
mands an  attraction  acting  accurately  in  accordance  with 
Newton's  law  (p.  103).  To  satisfy  the  law  of  areas,  it  is 
merely  necessary  that  the  attracting  force  be  what  is  called 
a  " central"  force,  directed  always  toward  a  definite  point 
occupied  by  the  sun  within  the  orbit.  And  conversely,  the 
fact  that  the  planets  can  be  observed  to  travel  in  orbits  that 
satisfy  the  law  of  areas,  proves  merely  that  they  are  moving 
under  the  influence  of  a  central  force,  but  not  necessarily 
that  particular  variety  of  central  force  which  we  know  under 
the  name  of  Newtonian  gravitation. 

But  in  addition  to  this  law  of  areas,  which  can  be  deduced 
as  a  fact  directly  from  observation  (p.  120),  two  other 
similar  laws  are  known,  —  also  obtainable  directly  from 
observation.  All  three  laws  were  first  found  by  Kepler; 
they  are  called,  to  the  present  day,  Kepler's  three  laws 
of  planetary  motion;  and  they  may  be  formulated  as 
follows : 

1.  The  orbit  of  each  planet  is  an  ellipse,  with  the  sun  at 
the  focus  of  the  curve. 

187 


ASTRONOMY 

2.  The  radius  vector  of  each  planet  passes  over  equal 
areas  in  equal  times. 

3.  If  the  time  required  by  any  planet  to  complete  a  revolu- 
tion in  its  orbit  is  called  its  " period,"  then  the  squares  of  the 
planetary  periods  are  proportional  to  the  cubes  of  their 
average  distances  from  the  sun.     This  third  law  is  called 
the  " harmonic  law."  * 

We  have  just  proved  that  the  second  law,  or  law  of  areas, 
is  a  necessary  consequence  of  the  existence  of  a  central 
force  pulling  always  toward  the  sun.  It  is  similarly  possible 
to  prove,  by  the  aid  of  mathematics,  that  all  three  laws 
follow  as  a  necessary  consequence  of  a  central  attracting 
force,  provided  that  force  acts  in  accordance  with  the 
Newtonian  law.  Thus  the  three  laws  of  Kepler  are  merely 
corollaries  or  consequences  of  Newton's  more  general  law ; 
Newton's  great  service  consisted  in  bringing  everything  under 
the  sway  of  a  single  law,  instead  of  three  separate  ones, 
apparently  unrelated.2 

In  the  light  of  the  above  explanation  of  Kepler's  and 
Newton's  work,  it  will  now  be  of  interest  to  give  a  brief 
account  of  the  two  best  known  explanations  of  planetary 
motion  within  the  solar  system,  —  the  Copernican  theory, 
which,  with  some  modifications,  is  the  one  now  accepted,  and 
the  older  Ptolemaic  theory.  It  may  possibly  seem  out  of 
place  to  give  any  attention  to  the  abandoned  Ptolemaic 
hypothesis;  it  is  like  studying  something  we  know  to  be 
untrue.  But  there  are  many  references  to  that  theory  in 

1  The  harmonic  law  may  be 'represented  mathematically  by  a  simple 
proportion : 

Let  h,  tz,  be  the  periods  of  two  planets,  a\,  a2,  their  mean  distances  from 
the  sun. 

Then  :  y  .  tj  =  Ols .  ajt 

2  Note  25,  Appendix. 

188 


THE   PLANETS 


Saturn 


literature  :  a  few  pages  may  well  be  devoted  to  a  description 
of  it ;  enough,  at  least,  to  form  some  idea  of  its  peculiarities. 
It  is  also  of  interest  that  the  Ptolemaic  theory  was  actually 
taught  in  early  days  at  Harvard  and  Yale  colleges,  as  being 
a  possible  alternative  theory  to 
the  Copernican.1 

Ptolemy  (140  A.D.),  following 
Hipparchus,  supposed  the  earth 
to  be  immobile,  near  the  center 
of  the  universe.  For  each  planet 
a  circular  orbit  was  provided 
(Fig.  50),  which  circle  was  called 
the  planet's  "  deferent."  Upon 
the  deferent  moved,  not  the 
planet  itself,  but  an  imaginary 
planet,  represented  by  a  point. 
The  actual  planet  moved  in 
another  circle  called  the  "  epi- 
cycle," whose  moving  center  was 
the  imaginary  planet.  The  sun 
and  moon  had  deferents,  but  no 
epicycles.  Each  deferent  was 
supposed  to  be  traced  on  the 
surface  of  a  perfectly  trans- 
parent separate  crystal  sphere ; 2 
and  all  these  crystal  spheres  rotated  once  a  day  around 
an  axis  passing  through  the  poles  of  the  heavens.  The 
outermost  crystal  sphere  had  no  deferent  or  attached 
epicycle ;  but  to  it  were  fastened  all  the  fixed  stars.  This 

1  Young,  Manual  of  Astronomy,  p.  323. 

2  These  spheres,  by  their  motion,  produced  the  famous  "music  of  the 
spheres." 

189 


FIG.  50.    Ptolemaic  Theory. 


ASTRONOMY 

star-sphere    also   rotated    around    the    polar   axis   of    the 
heavens. 

The  spheres  being  all  of  crystal,  and  perfectly  transparent, 
did  not  interfere  with  a  view  of  what  was  going  on  outside  of 
each  in  connection  with  the  exterior  deferents  and  epicycles. 
The  daily  axial  rotation  of  the  spheres  produced  all  the 
diurnal  phenomena  we  now  believe  to  result  from  the 
axial  rotation  of  our  earth.  And  the  spheres,  of  course, 
revolved  from  east  to  west,  not  as  our  earth  does,  from  west 
to  east. 

The  deferents  of  Mercury  and  Venus  were  inside  the  solar 
deferent.  The  imaginary  planets  Mercury  and  Venus 
revolved  in  their  deferents  once  a  year,  keeping  pace  with 
the  solar  motion  in  its  own  deferent  circle.  The  sun  and  the 
two  imaginary  planets  Mercury  and  Venus  were  always  in 
line,  as  shown  in  the  figure.  The  revolution  of  the  actual 
planets  Mercury  and  Venus  in  their  epicycles  thus  made 
them  swing  back  and  forth,  east  and  west  of  the  sun,  in  a 
manner  quite  similar  to  their  actual  observable  apparent 
motions  to  be  described  later  in  the  present  chapter. 

Mars,  Jupiter,  and  Saturn  were  connected  by  Ptolemy  to 
deferent  circles  exterior  to  the  sun.  The  periods  of  revolu- 
tion of  the  imaginary  planets  were  not  here  assumed  equal  to 
that  of  the  sun,  as  was  the  case  for  the  inferior  planets 
Mercury  and  Venus ;  and  in  this  way  the  observable  phe- 
nomena were  also  reproduced  for  these  superior  planets 
Later  investigators,  following  the  Ptolemaic  theory,  added 
further  secondary  imaginary  planets,  revolving  in  Ptolemy's 
epicyclic  circles;  with  the  actual  planets  attached  to  addi- 
tional corresponding  epicycles.  In  this  way  they  were  able 
to  reproduce  all  irregularities  of  motion,  as  improving 
methods  of  observing  brought  them  to  light. 

190 


THE   PLANETS 


In  contradistinction  to  the  above,  the  Copernican  theory, 
as  we  have  already  seen,  supposes  the  sun  immobile,  and  the 
planets  moving  in  flattened  oval  orbits  with  the  sun  at  one 
focus.  The  great  objection  to  this  system,  an  objection  that 
long  prevented  its  adoption  by  men  of  science,  is  this :  if 

the  earth  really 
revolves  in  an  orbit 
around  the  sun,  the 
fixed  stars  should 
change  their  ap- 
parent positions, 

as  seen  projected  on  the  sky,  while  the 
earth  progresses  around  its  orbit.  Figure 
51  makes  this  clear.  Let  S  be  the  sun; 
E'  and  E"  two  positions  of  the  earth  at 
opposite  points  of  its  orbit.  Suppose  a 
star  to  be  situated  in  space  at  P,  fixed  and 
immobile.  Then  from  E'  we  should  see 
the  star  projected  on  the  celestial  sphere 
at  P',  and  from  E"  we  should  see  it  at  P" . 
As  a  matter  of  fact  we  see  each  fixed  star 
constantly  projected  in  the  same  place  on 
the  celestial  sphere ;  and  this  seemed  an 
insuperable  objection  to  many  early  astron- 
omers, including  the  famous  Tycho  Brahe. 
On  the  other  hand,  if  the  earth  does  not 
move,  there  would  of  course  be  no  change 
in  the  direction  of  the  lines  E'P'  and  E"P".  There  would 
be  but  one  such  line  if  the  earth  were  constantly  in  the 
center  at  S. 

This  objection  to  the  Copernican  system  was  not  removed 
until  the  middle  of  the  nineteenth  century,  when,  for  the 

191 


FIG.  5i.     Ccp  ruioan 
System. 


ASTRONOMY 

first  time,  Bessel  was  able  to  measure  with  certainty  a  slight 
difference  between  the  two  sight  lines  from  the  earth  to  a 
certain  star  in  the  constellation  Cygnus.  It  then  appeared 
that  the  trouble  arises  from  the  extreme  minuteness  of  the 
angle  E'PE",  caused  by  the  fact  that  the  fixed  stars  are  all 
so  excessively  distant,  in  comparison  with  the  diameter  of  the 
earth's  orbit.  And,  of  course,  the  angle  E'PE"  will  diminish 
with  an  increasing  distance  of  the  stars.  Up  to  the  present 
time,  no  star  has  been  found  for  which  this  angle  exceeds  1.5 
seconds  of  arc ;  and  in  the  case  of  but  very  few  stars  has  the 
angle  been  found  large  enough  to  be  measured,  even  with 
the  powerful  astronomical  instruments  of  to-day.  The 
angle  subtended  at  P  by  the  radius  of  the  earth's  orbit  is  of 
course  half  the  angle  E'PE".  This  half-angle  is  called  the 
star's  " parallax."  And  the  measurement  of  even  a  single 
stellar  parallax  removes  the  fundamental  difficulty  of  the 
Copernican  theory. 

Of  historic  importance  even  greater  than  the  above 
theory  of  Ptolemy  are  certain  very  old  and  very  simple 
methods  of  determining  observationally  a  planet's  pe- 
riod of  revolution  around  the  sun  and  distance  from  the 
sun  in  terms  of  the  earth's  distance.  It  is  evident  that 
before  Kepler  discovered  his  harmonic  law,  no  relation  was 
known  to  exist  between  distance  and  period ;  but  there 
were  always  simple  methods  for  determining  the  period  by 
direct  observation.  When  we  were  discussing  the  earth  in  its 
relation  to  the  sun  (Chapter  VII),  we  found  that  the  great 
ecliptic  circle  on  the  sky  is  cut  out  by  the  plane  of  the  ter- 
restrial orbit  produced  outward  to  infinity.  It  must  also 
be  a  fact  that  any  planetary  orbit  plane  cuts  the  ecliptic  plane 
in  a  straight  line,  because  any  two  planes  in  space  must  in- 
tersect in  a  straight  line.  This  intersection  line  is  called 

192 


THE  PLANETS 

the  line  of  nodes  of  the  orbit.  Twice  in  the  course  of  its 
revolution  around  the  sun  the  planet  must  reach  this  line 
of  nodes.  When  this  occurs,  the  planet  is  for  a  moment  in 
the  ecliptic  plane  as  well  as  in  the  plane  of  its  own  orbit ; 
and  as  the  earth  is  always  in  the  ecliptic  plane  too,  it  follows 
that,  at  this  critical  moment,  the  straight  line  joining  the 
earth  and  planet  will  lie  entirely  in  the  ecliptic  plane. 

But  we  see  the  planet  along  that  line,  observing  from  the 
earth  toward  the  planet.  Consequently,  if  we  observe 
the  planet  at  the  critical  moment,  we  shall  see  it  projected 
on  the  sky  somewhere  in  the  great  circle  cut  out  on  the  sky 
by  the  ecliptic  plane.  So  we  can  ascertain  by  observation 
when  the  planet  is  in  the  node,  by  noting  the  instant  of  time 
when  it  crosses  the  ecliptic  circle,  as  seen  projected  on  the 
sky.  The  interval  between  two  successive  passages  of  the 
planet  through  the  same  node  is  then  its  period  of  revolution 
around  the  sun. 

Kepler  made  certain  important  improvements  in  the 
above  method  of  determining  planetary  periods;  and,  of 
course,  he  also  gave  much  time  to  the  study  of  planetary  dis- 
tances from  the  sun,  in  the  work  preparatory  to  his  discovery 
of  the  three  great  laws.  As  an  example  of  Kepler's  ingenious 
methods,  we  shall  give  here  his  investigation  of  the  varia- 
tions in  the  distance  between  the  earth  and  the  sun  in  dif- 
ferent parts  of  the  terrestrial  orbit.1  Kepler  had  at  his 

1  Kepler's  works  are  in  Latin,  and  are  difficult  to  read.  The  original 
book  from  which  we  quote  in  modernized  form  is  called  "  Astronomia  Nova 
sen  physica  coelistis  tradita  commentariis  de  motibus  stellae  Martis  ex  obser- 
vationibus  Tychonis  Brake."  It  was  published  in  1609,  but  there  is  a 
reprint  by  Dr.  Charles  Frisch,  published  in  1860  "Frankofurti  et  Er- 
langae." 

A  most  excellent  commentary  on  Kepler  was  also  published  in  London 
in  1804  by  the  Reverend  Dr.  Robert  Small,  and  dedicated  to  the  Earl  of 
Lauderdale. 

o  193 


ASTRONOMY 

disposal  a  long  series  of  observations  of  the  planet  Mars, 
accumulated  by  his  master  Tycho  Brahe.  These  observa- 
tions recorded  the  positions  of  Mars  as  seen  projected  on  the 
sky  on  a  very  large  number  of  different  dates.  He  selected 
certain  of  these  observations  dated  as  follows : l 

1590,  March  5,  7h  10m, 

1592,  Jan.  21,  6h  41m, 

1593,  Dec.  8,  6h  12m, 
1595,  Oct.  26,  5h  44m. 

It  will  be  seen  that  he  had  been  able  to  choose  four  ob- 
servations separated  by  exactly  the  same  interval  of  time; 

viz. :  686d  23h  31m,  which 
interval  corresponds  very 
nearly  with  the  known 
average  period  in  which 
>M  Mars  completes  a  revolu- 
tion around  the  sun.  In 
the  accompanying  Fig. 
52,  therefore,  Mars  must 
occupy  the  same  position 

FIG.  52.    Kepler's  Mars  Observations. 

M  on  each  of  the  above 

dates,  while  the  earth  will  occupy  the  successive  positions 
E,  F,  G,  H.  These  terrestrial  positions  will  be  equidistant 
points  on  a  circle  with  its  center  at  B,  if  we  suppose 
that  the  earth  moves  uniformly  in  a  circular  orbit.  Under 
this  supposition,  these  points  must  be  equidistant,  since 
they  are  separated  by  a  series  of  equal  time-intervals, 
each  equal  to  the  Martian  period.  And  it  is  important  to 
notice  that  the  successive  returns  of  Mars  to  the  same  point 
M  are  independent  of  any  assumption  as  to  the  form  or 

1  Frisch,  Kepler,  Vol.  3,  p.  275 ;   Small,  Kepler,  p.  202. 
194 


THE  PLANETS 

position  of  the  Martian  orbit.  Whatever  and  wherever  this 
orbit  may  be,  Mars  must  return  to  the  same  point  after 
each  complete  orbital  revolution  has  been  terminated. 

For  the  date  1590,  Mar.  5,  when  the  earth  was  at  E,  Kepler 
had  Tycho's  observation  of  the  position  of  Mars  as  pro- 
jected on  the  ecliptic  circle,  or  rather  the  position  of  that 
point  on  the  ecliptic  circle  which  was  nearest  to  Mars.  This 
gave  the  direction  of  the  sight-line  EM  from  the  earth  to 
Mars.  The  directions  of  the  lines  from  the  center  B  to  the 
earth,  and  from  the  center  to  Mars,  were  furnished  by  the 
tables  of  planetary  motion  in  Tycho's  possession.  Thus 
the  directions  of  the  three  sides  of  the  triangle  EBM  were 
known,  and  from  these  the  three  angles  of  the  triangle  were 
obtained  by  subtraction. 

But  when  the  three  angles  of  a  triangle  are  known,  it, is 
possible  to  calculate  the  relative  lengths  of  the  triangle's 
sides.1  By  successive  applications  of  this  process,  Kepler 
computed  that  2  — 

BE  =  .66774  X  BM. 
BF  =  .67467  X  BM. 
BG  =  .67794  X  BM. 
BH  =  .67478  X  BM. 

These  numbers  should  all  be  equal  if  the  point  around 
which  the  earth  describes  equal  angles  in  equal  times  were  at 
B,  the  center  of  the  circle  in  which  the  earth  is  supposed  to 
move.  So  Kepler's  numbers  show  that  the  point  about 
which  the  earth's  angular  motion  is  uniform  is  not  at  the 
center  B  of  the  earth's  orbit,  supposed  circular ;  but  that  it  is 
at  some  point  C,  outside  the  center  of  the  orbit.  Kepler  was 

1  In  the  language  of  trigonometry,  the  sides  are  proportional  to  the 
sines  of  the  opposite  angles. 

2  Frisch,  Kepler,  Vol.  3,  p.  275. 

195 


ASTRONOMY 

able  to  compute  the  position  of  this  point  C ;  and  the  corre- 
sponding changing  distance  between  the  earth  and  the  sun. 
These  results  were  of  course  obtained  long  before  he  perfected 
his  three  laws ;  they  are  regarded  justly  as  marking  one  of 
the  most  difficult  and  important  advances  ever  made  in 
human  knowledge. 

There  is  still  another  remarkable  peculiarity  about  the 
planetary  distances  from  the  sun;  like  the  foregoing,  of 
historic  interest  only.  When  we  compare  these  distances, 
we  find  an  accidental  relation  between  them.  Let  us  number 
the  planets  consecutively,  from  the  sun  outward,  calling 
Mercury,  1 ;  Venus,  2 ;  Earth,  3  ;  Mars,  4 ;  the  Planetoids, 
5 ;  Jupiter,  6 ;  Saturn,  7 ;  Uranus,  8 ;  Neptune,  9.  Let  us 
then  multiply  the  number  2  by  itself  four  times,  say  for 
Mars,  which  is  planet  number  4.  This  gives  16.  Then 
ta'ke  three-quarters  of  this  number,  giving  12.  Increase 
this  result  by  4,  giving  16.  Divide  this  by  10,  giving  1.6. 
The  result  is  an  approximate  value  for  the  distance  of 
Mars  from  the  sun,  counting  the  earth's  distance  from  the 
sun  as  1. 

This  curious  arbitrary  rule  is  known  as  Bode's  law ; 
astronomers  have  been  acquainted  with  it  for  more  than  a 
century ;  but  we  know  of  no  physical  reason  why  it  should 
have  a  real  existence.  The  following  little  table  contains  a 
comparison  of  the  known  planetary  distances  with  their 
values  calculated  as  above.  In  the  case  of  the  planetoids 
an  average  value  is  given. 

The  table  shows  that  the  law  is  quite  accurate  until  we 
reach  Neptune ;  then  the  error  increases  suddenly ;  and  we 
must  conclude  that  the  whole  thing  is  one  of  those  rare  and 
jremarkable  Coincidences  that  nature  sometimes  provides, 
apparently  to  mislead  scientific  investigators. 

196 


THE   PLANETS 


PLANET 

No. 

KNOWN 
DISTANCE 

"BODE" 
DISTANCE 

ERROR 
"  BODE  " 

Mercury  

1 

0.4 

0.5 

0.1 

Venus       

2 

0.7 

0.7 

0.0 

Earth  

3 

1.0 

1.0 

0.0 

Mars   

4 

1.5 

1.6 

0.1 

Planetoids    .... 
Jupiter     

5 
6 

2.6 
5.2 

2.8 
5.2 

0.2 
0.0 

Saturn      ...... 

7 

9.5 

10.0 

0.5 

Uranus 

8 

19.2 

196 

04 

Neptune 

9 

30.0 

38.8 

8.8 

Having  thus  described  certain  famous  historic  methods 
of  studying  the  planetary  distances,  etc.,  we  shall  next 
give  a  somewhat  more  detailed  description  of  the  planetary 
orbits,  and  the  exact  nature  of  the  observations  by  means 
of  which  we  study  them  in  modern  times.  When  we  deter- 
mine the  position  of  a  planet  by  observation,  we  really  deter- 
mine only  the  direction  in  which  we  see  it  projected  on 
the  celestial  sphere.  We  point  a  telescope  at  the  planet, 
and,  by  moving  the  telescope,  bring  the  center  of  the  plane- 
tary disk  very  accurately  into  the  middle  point  of  the  field 
of  view,  which,  for  this  purpose,  may  be  supposed  to  be 
fitted  with  a  very  fine  pair  of  cross  threads  to  mark  the  center. 
Then,  if  the  telescope  mounting  be  provided  with  suitable 
" graduated"  1  circles,  we  can  read  the  angles  measured  by 
those  circles,  and  thus  ascertain  the  direction  of  the  planet 
in  space,  referred  to  certain  points  and  lines,  such  as  the 
celestial  poles  and  equator.  In  other  words,  we  measure 
the  planet's  right-ascension  and  declination  (p.  34),  as  it  is 
seen  projected  on  the  celestial  sphere. 

We  can  also  note  the  exact  time  when  this  observation 

1  Brass  circles  divided  into  degrees,  minutes,  and  seconds  of  arc. 

197 


ASTRONOMY 

was  made,  thus  fixing  the  moment  when  the  planet's  direc- 
tion from  the  earth  was  measured.  There  are  other  methods 
of  making  these  observations  in  addition  to  direct  measure- 
ment with  graduated  circles  attached  to  the  telescope ;  but 
all  are  alike  in  this :  they  furnish  us  with  the  direction  in 
space  of  the  sight-line  joining  the  earth  with  the  planet, 
and  the  instant  of  time  when  that  line  had  the  direction 
in  question.  Direct  observation  gives  no  information 
whatever  as  to  the  planet's  distance  from  the  earth.  It  tells 
us  nothing  about  the  length  of  the  line  joining  earth  and 
planet ;  only  its  direction  in  space. 

If  several  observations  of  this  kind  have  been  made  at 
different  times,  separated  perhaps  by  a  number  of  days, 
or  even  months,  the  earth  will  itself  have  moved  consider- 
ably in  its  own  orbit  in  the  interval  between  the  observa- 
tions. The  planet  will  also  have  moved  in  its  own  orbit. 
Consequently,  both  ends  of  the  line  will  have  moved  in 
different  orbits  and  with  different  velocities;  so  that  the 
changes  in  direction  of  the  line  will  have  been  of  an  extremely 
complicated  nature. 

But  the  changes  in  space  of  one  end  of  the  line  are  well 
known  to  us,  —  the  earth  end.  For  we  know  the  orbit  of 
the  earth  around  the  sun,  and  can  calculate  the  terrestrial 
position  in  space  accurately  for  each  moment  of  time  when 
an  observation  was  made.  Knowing  thus,  from  calculation, 
the  position  of  one  end  of  the  line,  and,  from  observation 
the  direction  of  the  line,  the  line  itself  becomes  fully  known, 
all  but  its  length.  Thus,  in  Fig.  53,  if  at  a  certain  time  t\ 
the  earth  was  at  a  known  point  of  its  orbit  EI,  and  the 
planet  was  seen  in  the  observed  direction  PI,  we  know  the 
line  EiPi,  all  but  its  length.  If,  at  a  second  observation, 
made  at  the  time  t2,  the  earth  was  at  Ez,  and  the  planet 

198 


THE   PLANETS 

was  seen  in  the  direction  PZ)  we  again  know  the  line  E^Pi, 
excepting  its  length.  And  the  same  is  true  of  a  third  line 
E3PZ.  But  it  is  to  be  remarked  that  these  three  lines 
will  not  lie  in  a  single  plane,  unless  the  terrestrial  and  plane- 
tary orbits  around  the  sun  should  happen  to  lie  in  the  same 
plane,  which  is  not  accurately  the  case  in  our  solar  system. 

The  problem  now  is  to  determine  the  planetary   orbit 
from   observations   of   this   kind.     But   we   know   certain 
additional   things  about  this  planetary  orbit.     We  know 
that  it  is  an  ellipse  or  oval ;   that  the 
sun  is  in  the  focus;  and  we  know  the 
position  of  the  sun  with  respect  to  the 
earth  from  our  knowledge  of  the  terres- 
trial orbit,  since  the  sun  is  also  in  the 
focus  of  that  orbit.     Both  orbits  have 
the  same  point  for  a  focus,  and  the  sun 
is    in    that    point.      Furthermore,    we 
know  the  planet's  orbital  motion  must 
be  such  as  to  satisfy  Kepler's  laws  of     FlG"  53  va^0annset  Obser- 
planetary  motion  (p.  187),  and  so  we 
know  the  planet  must  have  moved  in  such  a  way  as  to 
cause  its  radius  vector  to  sweep  over  areas  proportional 
to  the  known  time-intervals  between  the  observations. 

It  is  a  fact  that  an  unknown  planetary  orbit  can  be  thus 
determined  from  three  observations  such  as  have  been  de- 
scribed. Our  geometric  problem  may  therefore  be  stated 
thus : 

Given  three  observed  straight  lines  in  space ;  it  is  re- 
quired to  find  an  ellipse,  cutting  these  three  lines  in  three 
points,  such  that  the  radii  vectores  to  the  sun  or  focus  from 
these  three  points  will  satisfy  Kepler's  laws. 

It  would  carry  us  too  far  afield  in  mathematical  astronomy 

199 


ASTRONOMY 

to  deduce  here  the  methods  by  which  this  problem  can  be 
solved;  but  several  interesting  things  about  it  can  be 
enumerated.  In  the  first  place,  the  problem  always  has 
two  solutions :  there  are  always  two  ellipses  in  space  that 
satisfy  the  problem.  One  of  these  is  the  planetary  orbit; 
the  other  is  the  earth's  own  orbit.  For  the  latter  is  also 
an  ellipse ;  it  cuts  the  three  lines  because  they  are  sight 
lines  from  the  earth  to  the  planet ;  and  the  earth's  motions 
in  its  own  elliptic  orbit,  of  course,  satisfy  Kepler's  law. 

But  suppose  the  problem  to  have  been  solved  for  the 
planet,  also ;  let  us  see  what  we  need  to  know  about  the 
orbit  in  order  to  say  that  we  know  the  orbit  completely. 
Six  different  things  must  become  known,  and  six  only; 
these  are  called  the  six  " elements"  of  a  planetary  orbit. 

First  we  must  know  the  length  of  the  largest  diameter 
of  the  ellipse,  and  the  degree  of  flattening,  —  the  eccentricity, 
as  it  is  called.  These  two  elements  being  known,  we  know 
the  size  and  shape  of  the  orbit.  We  could  draw  it  to  scale. 

Next  we  must  know  two  more  things,  to  define  where  the 
orbit  is  located  in  space.  These  two  elements  fix  or  deter- 
mine the  position  in  space  of  the  plane  in  which  the  orbit 
lies.  To  fix  this  plane,  we  must  know  the  angle  it  makes 
with  the  plane  of  the  earth's  orbit,  the  ecliptic  plane ;  and 
we  must  know  the  position  in  the  ecliptic  plane  of  the  line 
along  which  the  planetary  orbit  plane  cuts  the  ecliptic 
plane.  This,  as  we  have  seen,  is  called  the  "line  of  nodes"  ; 
and  the  angle  between  the  two  planes  is  called  the  "  inclina- 
tion" of  the  planet's  orbit. 

Having  thus  fixed  the  size  and  shape  of  the  orbit  in  its 
plane,  and  the  position  of  the  plane  itself  in  space,  we  must 
still  know  two  more  elements.  We  must  know  where  the 
planet  was  in  its  orbit  at  some  definite  time ;  and  we  must 

200 


THE  PLANETS 

know  the  position  of  the  orbit  in  its  own  plane.  As  we 
have  already  seen  (p.  120)  the  planet  is  said  to  be  in  peri- 
helion when  it  is  so  placed  in  its  orbit  as  to  be  at  its  nearest 
possible  approach  to  the  sun.  The  perihelion  point  is  that 
point  of  the  orbit  which  is  nearest  the  sun.  Therefore  we 
use  for  one  of  the  orbital  elements  the  exact  time  of  peri- 
helion passage.  This  element  fixes  the  position  in  the  orbit 
occupied  by  the  planet  at  a  definite  moment  of  time.  Finally, 
to  locate  the  orbit  in  its  own  plane,  we  must  know  the 
direction  in  that  plane  of  the  " major  axis,"  or  longest 
diameter  of  the  oval  orbit. 

The  six  elements  of  a  planetary  orbit  are  therefore  the 
following : 

1.  Longest  diameter  of  oval  1  These  fix  the  size  and  shape  of 

2.  Eccentricity,  or  degree  of  flattening  J      the  orbit. 

3.  Inclination  of  orbit  plane  1  These  fix  the  position  in  space  of  the  orbit 

4.  Position  of  lines  of  nodes  ]      plane. 

5.  Time  of  perihelion  passage. 

6.  Direction  of  orbital  major  axis  in  its  own  plane. 

A  seventh  orbital  element  is  usually  added :  the  Period, 
or  time  required  for  a  complete  orbital  revolution  of  the 
planet.  But  this  element  is  not  really  an  independent  one ; 
for  the  planetary  periods  and  the  diameters  of  the  orbits 
are  connected  by  Kepler's  harmonic  law  (p.  188),  by  means 
of  which  either  may  be  calculated  from  the  other. 

The  elements  of  an  orbit  once  computed  from  three  com- 
plete observations  of  the  planet's  apparent  position,  as  pro- 
jected on  the  celestial  sphere,  and  seen  from  the  earth,  the 
problem  can  be  inverted,  and  the  subsequent  apparent  pro- 
jected positions  of  the  planet  calculated  from  the  elements. 
Thus  is  it  possible  to  predict  exactly  where  each  planet  may 
be  seen  in  the  sky.  If  a  series  of  such  calculated  predictions 

201 


ASTRONOMY 

are  tabulated  for  every  day  in  the  year,  the  tabulation  is 
called  a  planetary  Ephemeris.  The  United  States  govern- 
ment publishes  such  a  tabulation  annually,  under  the  title 
"The  American  Ephemeris  and  Nautical  Almanac."  In  it 
the  planetary  positions  are  printed  for  each  day  in  the 
form  of  right-ascensions  and  declinations;  and  by  means 
of  these  printed  numbers  it  is  easy  to  find  the  planets  in 
the  sky. 

The  measurement  of  a  planet's  axial  rotation  period,  cor- 
responding to  the  terrestrial  sidereal  day,  is  not  a  very  easy 
matter.  The  best  method  of  doing  it  is  to  observe  with 
the  telescope  any  spot  or  mark  that  may  be  distinctly  visible 
on  the  planet's  surface.  As  the  planet  turns  on  its  axis,  this 
spot  will  alternately  appear  and  disappear;  for  it  will  of 
course  be  invisible  when  the  planet's  rotation  carries  it 
around  to  the  side  which  is  turned  away  from  our  earth. 
If  we  note  the  exact  time  elapsing  between  two  successive 
returns  of  the  spot  to  the  apparent  center  of  the  planet's 
disk,  this  interval  will  be  the  planet's  rotation  period,  or  day. 

Such  an  observation  must,  of  course,  be  corrected  for  any 
effects  produced  by  variations  in  the  relative  positions  of  the 
planet  and  the  earth,  due  to  their  respective  orbital  motions. 
And  the  result  can  also  be  much  improved  by  allowing  a 
considerable  number  of  rotations  to  elapse  between  the  two 
observations.  If  this  can  be  done,  the  effect  of  any  error 
in  noting  the  exact  time  when  the  spot  arrives  at  the  center 
of  the  disk  will  be  greatly  diminished.  But  none  of  the 
planets,  with  the  exception  of  Mars,  have  spots  sufficiently 
perfect  to  admit  of  precise  observation.  Our  knowledge  as 
to  the  duration  of  the  planetary  days  is  therefore  still  very 
defective. 

If  the  paths  of  the  spots,  as  they  move  across  the  visible 

202 


THE   PLANETS 

planetary  disk,  can  be  mapped  with  sufficient  accuracy,  we 
can  further  ascertain  from  them  the  location  of  the  plane- 
tary rotation  poles,  the  inclination  of  the  planetary  equator 
to  the  plane  of  the  orbit,  and  other  related  matters.  Unfor- 
tunately, information  of  this  kind  is  still  very  meager,  on 
account  of  the  lack  of  suitable  spots  on  the  surfaces  of  most 
planets. 

We  shall  next  consider  the  measurement  of  a  planet's 
size,  its  diameter,  surface  area,  and  volume.  We  have  seen 
that  ordinary  astronomical  instruments  enable  us  to  measure 


FIG.  54.    Planetary  Diameter. 


only  the  directions  in  which  we  observe  the  heavenly  bodies 
projected  on  the  celestial  sphere.  Thus,  for  instance,  we 
can  determine  whether  a  star  lies  in  the  direction  of  the 
celestial  equator,  or  whether  its  direction  makes  an  angle 
of  10°  with  the  direction  of  the  celestial  equator.  If  the 
former,  the  declination  (p.  34)  of  the  star  would  be  0° ;  if 
the  latter,  10°. 

Now  if  we  thus  observe  the  difference  in  direction  of  the 
two  sides  of  a  planetary  disk  (pp.  13,  52),  we  have  at  once 
the  " angular  diameter,"  or  the  angle  subtended  by  the 
planet  to  an  observer  on  the  earth.  Figure  54  explains  this 
matter.  E  is  the  observer  on  the  earth,  P  the  disk  of  the 
planet.  The  two  arrows  show  the  directions  in  which  the 
observer  sees  the  two  sides  of  the  planetary  disk  projected 

203 


ASTRONOMY 

on  the  celestial  sphere.  The  small  angle  at  E  is  the  differ- 
ence of  these  two  directions,  and  it  is  the  angular  diameter 
of  the  planet,  which  is  measured  by  observation. 

A  knowledge  of  this  angular  diameter  tells  us  nothing 
about  the  actual  diameter  of  the  planet  in  miles,  unless  we 
know  also  the  distance  D  between  the  earth  and  the  planet. 
For  it  is  obvious  that  it  would  require  twice  as  big  a  planet 
to  subtend  the  angular  diameter  observed  at  E,  if  the  planet 
were  removed  to  double  the  distance  D.  But  the  distance  Z), 
at  the  moment  of  observation,  can  always  be  calculated,  if 
we  know  the  dimensions  and  other  particulars  of  the  orbits 
pursued  by  the  earth  and  the  planet  around  the  sun.  And 
with  the  distance  D  available,  it  is  easy  to  calculate  the 
planet's  diameter  in  miles  from  the  observed  angular  diam- 
eter.1 

Having  thus  found  the  planet's  diameter  in  miles,  it  is 
frequently  convenient  to  represent  it  in  terms  of  the  earth's 
diameter  as  a  unit.  We  can  then  find  the  surface  area  of 
the  planet,  as  compared  with  that  of  the  earth,  by  simply 
squaring  the  planet's  diameter  expressed  in  terms  of  the 
earth's  diameter  as  unity.  And  the  same  number  cubed 
will  give  us  the  planet's  volume,  as  compared  with  the 
earth's.  For  it  is  a  well-known  mathematical  principle  that 
the  areas  of  spherical  bodies  are  proportional  to  the  squares 
of  their  diameters ;  and  their  volumes  are  proportional  to 
the  cubes  of  the  diameters. 

A  somewhat  more  difficult  problem  is  the  determination 
of  a  planet's  mass.  If  there  happens  to  be  a  satellite  re- 
volving around  the  planet,  the  problem  is  comparatively 

1  This  involves  merely  a  trigonometric  solution  of  the  long,  narrow 
triangle  shown  in  Fig.  54,  using  the  angle  at  E,  which  has  been  measured, 
and  the  two  including  sides,  which  are  both  equal  to  D  in  length. 

204 


THE  PLANETS 

easy.  We  can  then  determine  by  observation  the  period  of 
the  satellite's  revolution  in  its  orbit  around  the  planet ;  and 
its  distance  from  the  planet  in  miles  can  also  be  observed 
by  precisely  the  same  process  just  used  to  ascertain  the 
planet's  own  diameter  in  miles.  From  these  data  the 
planet's  mass  can  be  computed.1 

With  regard  to  the  planet's  satellites  in  general,  there  is 
not  much  more  to  be  said.  Their  distances  from  the 
planets  are  determined,  as  we  have  just  seen,  by  means  of 
angular  measures.  Their  periods  of  revolution  around  the 
planets  are  best  found  by  noting  the  time  elapsing  between 
successive  " elongations,"  or  occasions  when  the  satellite's 
orbital  motion  around  its  planet  carries  it  to  its  greatest 
apparent  angular  distance  from  the  planet. 

Most  satellite  orbits  are  almost  exact  circles :  our  own 
moon  has  an  exceptionally  flattened  or  elliptic  one.  And 
the  planes  of  the  satellite  orbits  are  mostly  very  near  the 
planes  of  the  planets'  equators ;  indeed,  the  equatorial  bulg- 
ing of  the  planet  itself  should  suffice  to  pull  the  orbit  plane 
of  a  close  satellite  into  the  planetary  equatorial  plane,  from 
gravitational  causes  alone.  That  the  planets  have  an  in- 
creased diameter  at  the  equator,  and  a  corresponding  polar 
flattening,  has  been  verified  by  direct  measurements  in  the 
case  of  our  earth  (p.  97).  For  the  other  planets  its  exist- 
ence is  proved  by  comparing  separate  determinations  of 
polar  and  equatorial  angular  diameters,  if  the  position  of 
the  poles  has  become  known.  When  the  satellites  are  un- 
usually far  from  their  planets,  as  in  the  case  of  our  moon, 
their  orbits  lie  nearly  in  the  planes  of  the  planets'  own 
orbits  around  the  sun. 

Before  leaving  this  subject  of  orbits  in  the  solar  system, 
1  Note  26,  Appendix. 
205 


ASTRONOMY 

we  shall  discuss  briefly  the  permanence  or  " stability"  of 
those  orbits.  Will  they  endure  forever?  Will  the  solar 
system  change  materially  in  the  course  of  time  ? 

The  planets  move  primarily  under  the  influence  of  solar 
attraction  as  if  they  were  themselves  mere  particles  devoid 
of  more  than  an  infinitesimal  mass.  They  are,  in  fact,  all 
extremely  small  in  comparison  with  the  great  sun.  Never- 
theless, they  do  possess  mass  in  a  certain  degree ;  and  con- 
sequently there  is  an  interaction  between  them,  which 
shows  itself  in  slight  perturbative  effects  upon  the  planetary 
orbits.  In  other  words,  if,  by  any  method,  we  determine 
the  elements  of  a  planetary  orbit  in  any  given  year,  we  shall 
not  find  these  elements  remaining  unchanged  forever. 
After  the  lapse  of  sufficient  centuries,  the  planetary  inter- 
actions and  perturbations  effect  changes  in  the  orbital  ele- 
ments of  the  solar  system. 

These  changes  are  of  two  kinds : 

1.  The  Periodic  perturbations. 

2.  The  Secular  perturbations. 

The  periodic  perturbations  increase  and  diminish  in  com- 
paratively brief  intervals  of  time,  comparable  in  length  to 
the  orbital  periods  of  the  planets  themselves.  But  the 
secular  changes,  produced,  as  it  were,  in  each  orbit  by  all 
the  other  orbits  acting  upon  it,  are  extremely  slow  in  period, 
requiring  many  thousands  of  years  to  complete  a  cycle. 

The  periodic  perturbations  never  displace  the  position 
in  which  we  see  a  planet  projected  on  the  celestial  sphere 
more  than  about  one  or  two  minutes  of  arc,  except  in  the 
case  of  Jupiter  and  Saturn,  which  are  at  times  displaced 
from  their  proper  or  unperturbed  orbital  positions  as  much 
as  half  a  degree,  more  or  less. 

The  most  interesting  facts  about  the  secular  perturbations, 

206 


THE  PLANETS 

known  from  the  researches  of  Laplace  and  Lagrange,  are  as 
follows : 

1.  The  major  diameters  and  periods  of  the  orbits  do  not 
change. 

2.  The  inclinations  and  eccentricities  vary  in  an  oscilla- 
tory manner. 

3.  The  nodal  points  and  perihelion  points  move  around 
the  ecliptic  and  orbital  planes,  respectively. 

4.  All  changes  of  whatever  kind  are  probably  oscillatory ; 
so  that  the  solar  system  is  stable  and  permanent.     After 
the  lapse  of  sufficient  ages,  it  will  always  return  again  to  its 
original  condition,  no  matter  what  changes  it  may  have 
undergone.     Of  this,  however,  there  exists  a  slight  doubt, 
due  to  a  possible  imperfection  discovered  recently  in  La- 
place's mathematical  demonstrations. 

5.  There  is  in  the  solar  system  an  "  in  variable  plane," 
not  subject  to  change,  and  containing  the  center  of  gravity 
of  all  the  bodies  composing  the  system. 

Throughout  the  foregoing  explanations,  the  word  " period" 
has  been  used  to  indicate  the  interval  of  time  required  by  a 
planet  to  complete  an  orbital  revolution  around  the  sun. 
But  there  exists  more  than  one  kind  of  planetary  period. 
When  we  were  discussing  the  planet  earth,  the  sidereal 
year  (p.  128)  was  defined  as  the  time  required  by  the  earth 
to  complete  one  orbital  revolution  around  the  sun.  Thus, 
if  we  imagine  an  observer  situated  on  the  sun,  the  sidereal 
year  will  be  the  time  elapsing  between  two  successive  ap- 
parent returns  of  the  earth  to  the  same  fixed  star,  if  both 
star  and  earth  are  supposed  to  be  seen  from  the  sun,  and 
projected  on  the  celestial  sphere.  In  the  same  way,  the 
sidereal  period  of  any  planet  is  the  time  required  for  a  com- 
plete orbital  revolution,  from  any  fixed  star  back  to  the 

207 


ASTRONOMY 

same  star,  and  seen  from  the  sun.  So  far  as  the  sidereal 
period  is  concerned,  then,  the  earth  is  in  precisely  the  same 
condition  as  all  the  other  planets. 

We  also  found  (p.  128)  that  the  earth  has  a  tropical  year, 
used  especially  in  calendar  making.  Of  course  no  other 
planet  has  a  tropical  year,  so  far  as  dwellers  on  the  earth 
are  concerned.  But  the  other  planets  all  have  another 
important  kind  of  year,  which  the  earth  does  not  have. 
It  is  called  the  Synodic  year  and  corresponds  to  the  synodic 
period  (p.  161)  in  the  case  of  the  moon.  To  define  it,  sup- 
pose, in  Fig.  55,  we  have  the 
orbits  of  the  earth  and  Jupiter. 
For  both  planets  the  sidereal 
year  is  the  time  required  to 
complete  revolutions  from  any 
two  points  such  as  E  and  J  back 
again  to  the  same  points.  But 
for  Jupiter,  which  has  a  synodic 
year,  this  synodic  year  is  defined 
as  beginning  when  a  straight 

FIG.  55.    Synodic  Year.  ,  «,t  ,      *i. 

line  drawn  from  the  earth  to  the 

sun  at  S  passes  through  Jupiter  at  J.  And,  similarly,  the 
synodic  year  ends  when  the  revolutions  of  both  bodies  make 
it  again  possible  to  draw  a  straight  line  from  the  earth  to 
Jupiter  through  the  sun. 

We  have  here  supposed  the  orbits  of  both  earth  and 
Jupiter  to  lie  in  a  single  plane.  This  may  be  done  as  a 
first  approximation  for  all  the  planets,  since  none  of  their 
orbits  lie  in  planes  very  greatly  inclined  to  the  ecliptic  plane, 
in  which  the  terrestrial  orbit  is  situated. 

Both  the  sun  and  Jupiter  are  seen  from  the  earth  pro- 
jected on  the  background  of  the  celestial  sphere;  conse- 

208 


THE  PLANETS 

quently,  when  they  are  in  this  straight-line  position,  they 
should  appear  to  us  at  the  same  point  on  the  sky.  Owing 
to  the  existing  small  angle  between  the  orbit  planes,  it  will 
happen  only  rarely  that  they  will  appear  to  occupy  the  same 
point  quite  exactly.  So  the  synodic  year  is  considered  to 
commence  when  they  are  as  nearly  as  possible  in  a  straight- 
line  position,  and  therefore  in  the  closest  possible  apparent 
proximity,  as  seen  by  us  projected  on  the  sky. 

At  such  a  time,  we  say  that  Jupiter  is  in  Conjunction 
with  the  sun.  In  general,  the  term  "  conjunction  "  is  thus 
used  whenever  two  celestial  bodies  are  in  very  close  prox- 
imity, as  seen  from  the  earth,  projected  on  the  celestial 
sphere. 

A  very  simple  mathematical  relation  exists  between  the 
synodic  and  sidereal  periods  of  any  planet.  It  is  based  on 
the  fact  that  the  synodic  period  depends  on  a  line  passing 
through  the  earth  as  well  as  the  planet,  and  must  therefore 
be  affected  by  the  terrestrial  as  well  as  the  planetary  rate 
of  orbital  motion ;  while  the  sidereal  period  depends  on  the 
planetary  motion  alone.1 

The  foregoing  reasoning  applies  strictly  to  those  planets 
only  whose  distances  from  the  sun  are  greater  than  that  of 
the  earth  from  the  sun.  These  are  called  Superior  planets 
to  distinguish  them  from  Mercury  and  Venus,  which  are 
accordingly  called  Inferior  planets,  because  their  orbits  lie 
within  that  of  the  earth. 

These  inferior  planets,  of  course,  have  sidereal  and  synodic 
periods  defined  in  the  same  way  as  the  corresponding 
periods  of  the  superior  planets.  The  accompanying  Fig.  56 
represents  the  case  of  an  inferior  planet  such  as  Venus. 
The  sidereal  period  of  Venus,  like  that  of  Jupiter,  is  the 

1  Note  27,  Appendix, 
p  209 


ASTRONOMY 

time  required  by  Venus  to  complete  an  orbital  revolution 
around  the  sun,  from  any  fixed  star  back  to  the  same  star 
again,  supposed  seen  from  the  sun.  But  when  we  draw  our 
straight  line  passing  through  the  sun,  the  earth,  and  Venus, 
Fig.  56  shows  that  such  a  line  can  be  drawn  when  Venus  is 
in  the  position  V,  or  in  the  position  V.  In  either  case, 
Venus  and  the  sun  will  be  seen  from  the  earth  close  together, 
as  projected  on  the  celestial  sphere ;  and  will  therefore  be 
in  conjunction.  When  Venus  is  thus  in  conjunction  through 
being  situated  between  the  sun  and  the  earth,  we  call  the 
phenomenon  Inferior  Conjunction  ;  and 
when  the  sun  is  between  Venus  and  the 
earth,  we  call  it  Superior  Conjunction. 

Of  course  a  superior  planet,  like  Jupiter, 
whose  orbit  is  entirely  outside  that  of  the 
earth,   can  never  be   placed    between   the 
FlG-  f$;m£ferior     earth  and  the  sun,  and  can  therefore  never 
have    an    inferior    conjunction.      Superior 
planets  have  superior  conjunctions  only;    inferior  planets 
have  both  inferior  and  superior  conjunctions. 

The  synodic  period  of  Venus  is,  then,  the  time  in  days 
elapsing  between  two  successive  inferior  conjunctions,  or 
two  successive  superior  conjunctions.  But  the  mathe- 
matical relation  connecting  the  synodic  and  sidereal  periods 
is  slightly  different  from  that  which  holds  in  the  case  of  a 
superior  planet.1 

The  following  little  table  contains  approximate  plane- 
tary periods ;  and  exhibits  the  interesting  fact  that  both 
kinds  of  periods  increase  from  Mercury  to  Mars,  inclu- 
sive. Also,  for  this  part  of  the  table,  the  synodic  periods 
are  always  the  greater  periods.  But  for  all  the  other 

1  Note  28,  Appendix. 
210 


THE   PLANETS 


SIDEREAL  PERIOD 

SYNODIC  PERIOD  1 

Mercury        .          ...          .     . 

88  days 

116  days 

Venus        .     »- 

225     " 

584     " 

Earth  -.•-.- 

365     " 

Mars    . 

687     "     '•' 

780     " 

Jupiter  .         '". 

12  years 

399     " 

Saturn  •  . 

30     " 

378     " 

Uranus      * 

84     " 

370     " 

Neptune        

165     " 

368     " 

planets  the  synodic  periods  are  far  smaller  than  the  si- 
dereal periods;  and  they  are  all  nearly  equal  in  dura- 
tion.2 

It  is  plain  that  when  any  planet  is  in  conjunction  with 
the  sun,  we  shall  be  unable  to  see  it.  Sun  and  planet  being 
then  projected  on  the  sky  at  nearly  the  same  point,  the 
bright  solar  light  will,  of  course,  overcome  the  faint  planet, 
and  make  it  invisible.  In  other  words,  the  planet,  appearing 
near  the  sun,  will  be  above  the  horizon  in  daytime.  To 
make  the  planet  visible,  it  must  be  far  from  the  sun,  as  seen 
projected  on  the  sky ;  i.e.  there  must  have  been  consider- 
able synodic  motion  since  the  time  of  conjunction.  Visi- 
bility from  the  earth  depends  on  synodic  motion,  not  actual 
motion  in  the  orbit. 

It  is  customary  to  use  the  term  " elongation"  to  designate 
a  planet's  angular  distance  from  the  sun,  as  we  see  it  pro- 
jected on  the  sky.  At  the  time  of  conjunction,  the  planet's 
elongation  is  very  small ;  it  may  even  be  zero.  We  have 
seen  in  Figs.  55  and  56  the  state  of  affairs  when  a  conjunc- 
tion with  the  sun  occurs  in  the  case  of  a  superior  and  inferior 

1  These  periods  have  been  used  on  pp.  50  and  51. 

2  Note  29,  Appendix. 

211 


ASTRONOMY 


planet.  As  the  synodic  motion  advances  after  conjunction, 
the  planets  increase  their  elongation  from  the  sun.  Figures 
57  and  58  show  the  maximum  elongations  the  two  kinds  of 

planets  can  attain.  For  the 
superior  planets,  like  /,  repre- 
senting Jupiter  (Fig.  57),  the 
elongation  may  reach  180°.  For 
the  inferior  planets,  like  V,  repre- 
senting Venus  (Fig.  58),  there  is 
a  certain  definite  maximum  angle 
of  elongation  SEV,  which  occurs 
when  there  is  a  right  angle  at 
V ;  i.e.  when  there  is  a  right  angle 

Fio.  57.    Superior  Planet.    Greatest     between    the   directions   of    earth 

and  sun,  as  seen  from  the  planet. 

When  the  elongation  is  180°  in  the  case  of  a  superior 
planet  (Fig.  57),  the  sun  is  directly  opposite  the  planet,  as 
seen  from  the  earth,  projected  on  the  sky.  Thus,  in  the 
figure,  the  sun  would  be  seen  from  the  earth  E  projected 
toward  the  upper  part  of  the  page,  and  Jupiter  directly 
opposite,  projected  toward  the  lower  part 
of  the  page.  The  planet  is  then  said  to  be 
in  Opposition.  The  greatest  possible  elon- 
gations (Fig.  58)  for  the  inferior  planets 
Mercury  and  Venus,  which  can  never  be  in 
opposition,  are  47°  1  for  Venus,  and  28°  for 
Mercury.  These  numbers  may  be  verified  by 
means  of  a  simple  mathematical  calculation.2 

Let  us  still  remember  that  for  the  purpose  of  a  first  ap- 
proximation we  may  consider  all  the  planetary  orbits  to  lie 
in  a  single  plane,  the  plane  of  the  ecliptic.  It  follows  that 

1  Cf.  p.  51.  2  Note  30,  Appendix. 

212 


FIG.  58.  Inferior 
Planet.  Greatest 
Elongation. 


THE   PLANETS 

we  must  always  see  the  planets  projected  on  the  sky  near 
the  great  circle  cut  out  by  that  plane, — the  ecliptic  circle, 
in  which  we  also  see  the  sun  projected.  Now  since  Mercury 
is  thus  always  near  the  ecliptic  circle,  and  always  within  28° 
of  the  sun,  it  must  appear  to  us  to  oscillate  back  and  forth 
near  the  ecliptic  circle,  appearing  now  on  one  side  of  the 
sun,  now  on  the  other.  This  is  also  true  of  Venus,  the 
other  inferior  planet,  though  here  the  arc  of  oscillation  is 
much  greater,  as  we  have  seen.  When  Mercury  is  at  either 
extreme  of  its  oscillation,  it  is  in  greatest  elongation.  When 
it  is  an  eastern  elongation,  Mercury  being  east  of  the  sun, 
the  planet  is  visible  for  a  short  time  after  sunset.  When 
it  is  a  western  elongation,  the  planet  is  west  of  the  sun,  and 
is  visible  a  short  time  before  sunrise.  But  owing  to  the 
apparent  proximity  of  the  sun,  Mercury  is  always  projected 
against  the  rather  bright  background  of  the  sky  near  the 
point  where  the  sun  rises  or  sets  at  the  horizon.  Thus 
Mercury  is  not  very  easy  to  see.  Venus,  with  its  much 
greater  possible  elongation  angle,  is  a  very  easy  object  to 
the  unaided  eye. 

In  general,  we  thus  find  that  the  visibility  of  an  inferior 
planet  depends  on  the  production  of  these  maxima  of  elonga- 
tion by  the  synodic  motion  (cf.  p.  50). 

In  the  case  of  a  superior  planet  the  state  of  affairs  is  very 
different.  Visibility  still  depends  on  synodic  motion ;  as 
before,  the  planets  cannot  be  seen  near  the  time  of  conjunc- 
tion. But  as  their  synodic  motion  advances,  these  planets 
do  not  approach  a  moderate  maximum  elongation,  and 
appear  to  oscillate  back  and  forth  across  the  sun.  For,  as 
we  have  already  seen,  the  superior  planets  have  their  oppo- 
sitions when  their  elongation  from  the  sun  is  180° ;  then 
they  are  directly  opposite  the  sun ;  and  are  therefore  observ- 

213 


ASTRONOMY 

able  on  the  visible  part  of  the  celestial  meridian  at  mid- 
night, when  the  sun  is  on  the  lower  and  invisible  part 
(cf.  p.  51). 

But,  nevertheless,  the  superior  planets  do  have  certain 
oscillations  in  their  apparent  motions  among  the  stars,  as 
seen  from  the  earth.  These  oscillations  cause  them  to  per- 
form at  times  so-called  " retrograde"  motions,  traveling 
apparently  among  the  stars  from  east  to  west  instead  of 
west  to  east,  which  is  their  usual  direction  of  apparent 
motion.  Sometimes,  too,  they  have  temporary  "  station- 
ary points/'  appearing  immobile  for  a  short 
time,  like  fixed  stars. 

To  understand  this  state  of  affairs,  let 
us  consider  for  a  moment  the  orbits  of  the 
earth  and  a  superior  planet  like  Mars.  The 
accompanying  Fig.  59  shows  these  orbits, 
not  drawn  to  scale,  but  again  supposed  to 
be  in  a  single  plane,  and  circular.  Be- 
FIG.  59.  Retrograde  ginning  at  the  time  of  opposition,  Mars, 

Motion  of  Mars.  .  ,  .  .       ..         -mmn 

earth,  and  sun  are  shown  on  the  line  MES. 
At  the  end  of  one  month,  Mars  will  be  at  M'  and  the  earth 
at  E'.  After  three  months,  Mars  will  be  at  M"  and  the 
earth  at  E".  At  these  three  dates,  therefore,  terrestrial 
observers  will  see  Mars  projected  on  the  sky  along  the  three 
successive  directions  EM,  E'M',  and  E"M".  Both  planets 
have  been  constantly  and  uniformly  moving  in  the  direction 
of  the  curved  arrows,  yet  from  EJ  we  see  Mars  along  E'M', 
apparently  retrograded  back  of  the  direction  EM,  or  contrary 
to  the  direction  of  orbital  motion  for  both  planets.  At 
E"M" ,  Mars  has  again  begun  to  move  forward  in  its  ap- 
parent motion  among  the  stars,  and  that  forward  motion 
will  evidently  become  more  rapid  a  little  later.  It  is  also 

214 


THE   PLANETS 

clear  that  about  the  time  the  apparent  motion  changes  from 
retrograde  to  direct,  Mars  will  for  a  short  time  appear 
quite  stationary  among  the  stars.  And  it  is  further  evident 
from  the  figure  that  the  middle  of  the  arc  of  apparent  retro- 
gression must  occur  about  the  time  of  opposition,  when  the 
planet  is  nearest  the  earth. 

There  is  but  one  more  peculiarity  of  importance  in  con- 
nection with  this  apparent  motion  of  the  planets  as  seen 
from  the  earth,  and  projected  on  the  sky.  It  arises  from 
the  fact  that  the  orbital  planes  do  not  coincide  accurately 
with  the  ecliptic  plane,  and  therefore  the  planets  do  not 
always  appear  to  us  on  the  sky  projected  accurately  on  the 
ecliptic  circle.  They  have  certain  small  apparent  motions 
toward  the  ecliptic  circle,  and  again  away  from  it.  It 
follows  that  a  planet's  arc  of  retrograde  motion  does  not 
simply  return  along  the  same  line  over  which  it  traveled 
in  its  direct  motion,  as  would  be  the  case  if  all  planetary 
motions  were  accurately  in  the  ecliptic  plane.  The  actual 
retrograde  apparent  motions  usually  involve  peculiar  curves, 
both  for  the  superior  and  inferior  planets. 

We  shall  close  this  chapter  with  another  reference  to  the 
Keplerian  method  of  determining  the  planetary  periods. 
The  matter  could  not  be  explained  fully  until  the  synodic 
period  had  been  made  clear.  By  the  aid  of  that  period, 
astronomers  of  old  possessed  still  another  simple  way  of 
ascertaining  the  sidereal  period  by  observation.  They  could 
observe  the  date  when  the  planet  was  in  opposition  to  the 
sun,  when  it  comes  to  the  meridian  at  midnight.  Then 
the  interval  between  two  successive  oppositions  is  the  synodic 
period  (p.  208) ;  and  from  the  synodic  period  they  could 
calculate  the  sidereal  period,  which  is  the  true  period  of 
orbital  revolution,  by  means  of  a  simple  mathematical 

215 


ASTRONOMY 

equation.1  This  method  cannot  be  used  for  the  inferior 
planets,  as  they  do  not  have  oppositions. 

The  accuracy  of  this  measurement  of  period  could  be 
increased  greatly  by  comparing  two  oppositions  between 
which  the  planet  had  made  many  revolutions  around  the 
sun.  Thus,  by  a  comparison  of  two  oppositions  separated 
by  five  hundred  synodic  periods,  the  error  of  observation 
affecting  the  exact  times  of  opposition  would,  of  course,  be 
divided  by  500.  This  was  actually  possible  in  the  case  of 
the  principal  planets,  by  utilizing  existing  ancient  records 
of  opposition  observations. 

Furthermore,  it  was  necessary  to  compare  distant  oppo- 
sitions, to  eliminate  the  effects  of  orbital  flattening  in  the 
case  of  both  the  planet  and  the  earth.  For  it  is  clear  that 
successive  synodic  periods  will  not  be  accurately  equal : 
they  would  be  so,  if  the  orbits  were  truly  circular ;  but  from 
the  average  of  a  large  number  of  successive  revolutions  this 
source  of  error  is  practically  removed. 

1  Note  27,  Appendix. 


216 


CHAPTER  XII 

THE    PLANETS   ONE   BY   ONE 

IT  will  now  be  of  interest  to  consider  separately  the  many 
details  in  which  the  planets  differ  amongst  themselves ;  and 
we  shall  begin  with  Mercury,  the  one  nearest  the  sun.  As 
we  know,  it  always  appears  projected  on  the  sky  in  the 
vicinity  of  the  sun  (p.  50) ;  sometimes  on  one  side  of  it, 
sometimes  on  the  other.  The  ancients  did  not  perceive 
that  this  planet,  seen  alternately  on  opposite  sides  of  the 
sun,  was  a  single  body.  They  had  two  names  for  it,  — 
Apollo  and  Mercury. 

The  seasons  on  Mercury  must  present  a  rather  curious 
problem.  We  have  no  means  of  ascertaining  with  any 
degree  of  certainty  the  angle  between  this  planet's  rotation 
axis  and  the  plane  of  its  orbit  (p.  203).  On  the  earth  this 
angle  is  66 J° ;  and  it  is  owing  to  the  existence  of  such  an 
angle  that  we  have  the  regular  terrestrial  seasons  (p.  120). 
Therefore  we  know  very  little  about  the  seasons  of  Mercury, 
so  far  as  they  may  be  analogous  to  terrestrial  seasons.  But 
we  know  that  the  distance  of  this  planet  from  the  sun  has 
so  large  a  variation  between  perihelion  and  aphelion  that 
a  very  variable  quantity  of  solar  heat  must  reach  it  at 
different  times.  There  must  exist  a  variability  from  this 
cause,  great  enough  to  make  very  appreciable  temperature 
changes.  The  interaction  of  this  with  possible  seasons  of 
the  terrestrial  kind  may  give  rise  to  hot  summers  and  cold 
summers,  etc.,  in  different  years. 

217 


ASTRONOMY 

In  the  telescope,  Mercury  exhibits  phases  like  our  moon, 
and  due  to  the  same  cause.  It  has  little  or  no  atmosphere 
in  all  probability ;  and  most  astronomers  can  see  but  the 
faintest  surface  markings.  Lowell,  however,  has  published 
drawings  of  Mercury  showing  many  geometric  lines  and 
angles ;  and  he  thinks  they  change  their  apparent  positions 
on  the  planet  very  slowly.  This  would  indicate  that  there 
is  no  rapid  axial  rotation  like  the  earth's.  If  the  planet 
turned  quickly  on  its  axis,  the  rotation  would  soon  carry 
some  of  the  markings  out  of  sight  around  the  edge  of  the 
planet  (p.  202).  Possibly,  therefore,  Mercury,  like  the  moon, 
rotates  on  its  axis  but  once,  while  making  an  orbital  circuit 
around  the  sun  in  88  days.  If  this  be  so,  there  must  be  a 
very  hot  hemisphere,  always  facing  the  sun,  and  a  very 
cold  opposite  hemisphere.  But  this  would  be  modified 
somewhat  by  the  very  large  librations  (p.  171),  which  would 
result  from  Mercury  having  an  unusually  flattened  orbit 
around  the  sun. 

The  surface  of  Mercury  is  not  very  brilliant.  It  has  been 
calculated  that  it  reflects  only  13  per  cent  of  the  solar  light 
falling  upon  it.  This  p^'^tflffp  nf  liflht-reflection  is  called 
the  planet's  Albedo ;  and  Mercury  has  the  lowest  albedo  in 
the  solar  system. 

The  planet  Venus,  the  other  inferior  planet,  is  also  seen 
alternately  on  opposite  sides  of  the  sun,  appearing  as  morning 
and  evening  star.  But  it  attains  a  much  greater  angular 
distance  from  the  sun  than  does  Mercury,  and  is  also  more 
brilliant.  It  is  at  times- the  brightest  of  all  the  planets, 
and  can  even  be  seen  by  the  unaided  eye  in  full  day- 
light near  the  occasions  of  its  greatest  elongation  from  the 
sun. 

The  telescopic  phases  of  Venus  range  all  the  way  from  a 

218 


FIG.  60.    Ptolemy's  Theory  of  Venus. 


THE   PLANETS  ONE  BY  ONE 

complete  circle  down  to  a  narrow  crescent.1  According  to  old 
Ptolemy's  theory  (p.  189),  we  should  never  see  the  phase  of 
Venus  larger  than  the  half- 
moon  shape.  For  Ptolemy 
supposed  Venus  moving  on 
a  circle  whose  center  was 
always  near  a  line  joining 
the  earth  and  the  sun.  It 
is  clear,  from  Fig.  60,  that 
the  angle  at  Venus  be- 
tween the  earth  and  the 
sun  could  never  be  as 
small  as  a  right  angle; 
and  so  Venus  could  never 
show  a  phase  bigger  than  the  half-moon,  according  to  the 
accepted  Aristotelian  theory  of  phase  phenomena  (p.  163). 

This  matter  is  most  in- 
teresting ;  the  moment 
Galileo  turned  the  first 
astronomic  telescope  upon 
Venus,  about  the  year 
1610,  and  saw  a  phase 
larger  than  the  half -moon, 
he  had  at  once  a  strong 
proof  that  something  was 
wrong  with  this  particular 
detail  of  the  sacrosanct 
Ptolemaic  theory.  To 

FIG.  61.    Greatest  Luminosity  of  Venus.  , 

remove     this     difficulty, 

however,  it  would  only  have  been  necessary  for  Ptolemy  to 
lengthen  the  radius  of  the  Venus  epicycle. 

1  Well  shown  in  Plate  8,  p.  225,  a  reproduced  photograph. 
219 


ASTRONOMY 

Venus  gives  a  good  example  to  demonstrate  that  a  planet 
does  not  attain  its  greatest  luminosity  when  nearest  the 
earth,  nor  when  exhibiting  the  largest  possible  phase.  Fig- 
ure 61  illustrates  this  problem.  The  points  S,  V,  E  repre- 
sent positions  of  the  sun,  Venus,  and  the  earth  at  inferior 
conjunction  (p.  210).  Venus  has  then  no  perceptibly  disk ; 
we  see  its  dark  side.  As  time  goes  on,  the  phase  of  Venus 
grows;  and  the  light  reflected  toward  us  increases  in  pro- 
portion to  the  increasing  area  of  the  visible  disk.  But 
at  the  same  time  the  distance  from  Venus  to  the  earth 
is  increasing;  and  the  intensity  of  the  planet's  light,  as 
received  by  the  earth,  of  course  diminishes  rapidly  with  the 
increase  of  our  distance  from  Venus.  Thus,  in  the  figure, 
at  the  moment  when  Venus  has  reached  the  point  V,  the 
earth  is  at  E'.  The  area  or  phase  of  the  visible  disk  has 
grown  from  zero  at  V  to  the  segment  shown  unshaded  at  V ', 
but  the  distance  between  the  two  planets  has  increased  from 
VE  to  V'E'. 

Thus  we  see  that,  beginning  with  inferior  conjunction, 
the  disk  area  grows  much  more  rapidly  than  the  distance ; 
consequently,  Venus  grows  more  brilliant  to  our  eyes. 
But,  later  on,  this  is  reversed ;  so  there  must  be  a  certain 
point  where  Venus  suddenly  begins  to  substitute  a  decrease 
of  visible  brilliancy  for  the  previous  increase.  This  is  the 
moment  of  maximum  luminosity,  as  seen  from  the  earth ; 
it  is  a  nice  problem,  requiring  the  infinitesimal  calculus  for 
its  solution,  to  determine  this  moment  exactly.  It  will 
suffice  here  to  say  that  it  occurs  about  36  days  from  inferior 
conjunction,  when  Venus  has  a  phase  like  the  crescent  moon. 

Venus  is  believed  to  possess  an  atmosphere,  for  it  has  a 
very  high  albedo,  or  light-reflecting  power,  which  indicates 
a  reflecting  surface  containing  clouds.  Moreover,  Venus  is 

220 


THE   PLANETS  ONE  BY  ONE 

occasionally  seen  to  pass  between  the  earth  and  the  sun, 
-a  phenomenon  called  a  Transit  of  Venus.  When  these 
transits  are  about  to  occur,  and  just  as  the  planet  is  be- 
ginning to  encroach  upon  the  solar  disk,  as  seen  from  the 
earth,  a  ring  of  light  becomes  visible  around  the  part  of 
Venus  not  yet  projected  upon  the  sun.  This  cannot  be  ex- 
plained otherwise  than  as  a  refraction  or  reflection  of  solar 
light  by  the  planetary  atmosphere. 

Certain  ill-defined  shadings  have  been  seen  at  times  on 
the  planet's  surface :  Lowell  goes  so  far  as  to  give  a  map  of 
Venus  showing  very  clear  geometrical  structures  of  straight 
lines.  These  are  of  interest  because  of  their  bearing  on 
Lowell's  observations  and  theories  as  to  the  Mars  " canals." 
From  observations  of  the  markings  he  concludes  that 
Venus  (as  he  also  found  in  the  case  of  Mercury)  has  a  very 
slow  axial  rotation ;  that  it  probably  turns  on  its  axis  in  225 
days,  which  is  also  its  sidereal  period.  If  this  be  correct, 
Venus  must  always  turn  the  same  face  toward  the  sun. 

The  planet  Mars,  which  we  shall  next  consider,  differs 
greatly  from  Mercury  and  Venus.  Its  orbit  is  exterior 
to  that  of  the  earth  and  varies  quite  considerably  from  an 
exact  circular  form,  so  that  the  planet's  distance  from  the 
sun,  and  its  distance  from  the  earth,  undergo  very  wide 
variations,  corresponding  to  the  planet's  motion  in  its  orbit. 
Furthermore,  unlike  Mercury  and  Venus,  Mars  has  certain 
very  well-defined  and  constantly  visible  surface  markings. 
These  have  enabled  astronomers  to  ascertain  with  precision 
the  length  of  the  Martian  day,  or  period  of  axial  rotation.  It 
is  found  to  be  about  24f  of  our  terrestrial  hours,  or  nearly 
the  same  as  the  day  of  our  earth.  The  diameter  of  Mars  is 
about  half  that  of  the  earth ;  and  the  inclination  of  its  axis 
to  the  plane  of  its  orbit  around  the  sun  is  65°.  Since  the 

221 


ASTRONOMY 

corresponding  angle  of  inclination  in  the  case  of  the  earth  is 
66f  °,  it  is  clear  that  the  Martian  seasons  will  resemble  closely 
those  experienced  by  ourselves.  Thus  there  are  many  points 
of  resemblance  between  the  two  planets,  Mars  and  the  earth ; 
and  therefore  is  Mars  the  best  hunting  ground  for  those  who 
seek  a  planet  with  intelligent  inhabitants. 

In  the  telescope,  Mars  shows  no  crescent  phases  like  the 
moon  or  the  inferior  planets,  because  its  orbit  is  outside 
that  of  the  earth;  and  so  the  angle  at  Mars  between  the 
earth  and  the  sun  can  never  be  as  big  as  a  right  angle.  Its 
atmosphere  should  be  less  dense  than  that  of  the  earth ; 
for  the  absence  of  clouds  is  indicated  by  our  seeing  con- 
stantly permanent  markings  on  the  planet's  own  surface, 
and  by  the  observable  fact  that  Mars  has  an  unusually  low 
albedo,  or  light-reflecting  power.  Moreover,  owing  to  its 
small  size  and  small  mass,  the  attractive  force  of  gravity  on 
Mars  is  less  than  half  that  existing  on  the  earth.  Conse- 
quently, it  is  not  improbable  that  Mars  has  been  deprived 
of  its  atmosphere  in  the  same  way  that  the  moon  is  believed 
to  have  lost  its  own  air  (p.  167). 

Mars  has  two  satellites  or  moons ;  and  they  are  in  some 
respects  the  most  peculiar  bodies  in  the  solar  system.  Their 
special  oddity  arises  from  their  close  proximity  to  the  planet, 
and  the  consequent  shortness  of  their  periods  of  orbital 
revolution  about  it.  Deimos,  the  outer  satellite,  has  an 
orbital  period  of  30h  18m.  Phobos,  the  inner  one,  revolves 
in  its  orbit  in  7h  39m.  These  brief  intervals  are  the  "  lunar 
sidereal  periods  "  (p.  161)  for  Mars.  Now  the  planet  itself 
takes  about  24f  hours  to  complete  an  axial  rotation.  There- 
fore the  orbital  motion  of  Phobos,  as  seen  from  Mars,  makes 
it  move  among  the  stars  from  west  to  east  much  faster  than 
the  apparent  diurnal  motion  of  the  Martian  celestial  sphere 

222 


THE   PLANETS  ONE   BY  ONE 

makes  the  satellite  seem  to  move  from  east  to  west.     In  other 
words,  Phobos  rises  in  the  west  and  sets  in  the  east ! 

Deimos,  however,  with  its  period  of  30h  18m,  travels  diur- 
nally  from  east  to  west.  We  can  investigate  easily  its 
apparent  diurnal  motion.  Its  period  being  30. 3h,  in  one 

360° 

hour  it  moves  among  the  stars In  the  same  time  the 

30.3 

360° 
diurnal  rotation  of  the  Martian  celestial  sphere  is  ^j-=-     The 

apparent  motion  of  Deimos  is  therefore  : 
360°     360° 


24.7      30.3 


in  an  hour,  east  to  west. 


It  will  therefore  make  an  apparent  rotation  of  360°  around 
the  sky  in  a  number  of  hours  found  by  dividing  360°,  the 
circumference  of  an  entire  diurnal  circle,  by  the  above 
hourly  motion.  The  result  of  this  division  is  128  hours; 
and  this  is  the  " lunar  day"  (p.  176)  of  Deimos.  And  so  we 
have  the  unusual  condition  that  the  lunar  day  is  far  longer 
than  the  lunar  "month,"  or  sidereal  period. 

Approaching  now  the  question  of  Martian  "inhabitants," 
and  their  canals,  we  must  first  inquire  as  to  the  existence  of 
water  vapor  in  the  atmosphere  of  the  planet.  Is  there 
any?  This  is  a  matter  of  prime  importance  in  connection 
with  the  famous  supposed  canals.  If  there  exists  on  the 
planet  a  network  of  geometric  markings,  their  explanation 
as  waterways  must  stand  or  fall  by  the  water  vapor  in  the 
planetary  air.  A  flow  of  water  in  canals  can  be  imagined 
only  if  we  suppose  also  evaporation  of  that  water  into  an 
atmosphere,  and  subsequent  precipitation  of  it  as  snow  or 
rain.  If  this  precipitation  occurs  for  some  reason  principally 
near  the  planet's  poles,  while  the  evaporation  takes  place 
all  along  the  canals,  we  might  imagine  the  latter  to  have 

223 


ASTRONOMY 

been  constructed  artificially  to  carry  the  water  away  from 
the  poles,  so  as  to  fructify  and  irrigate  the  entire  planetary 
surface. 

When  these  markings  were  first  seen  by  Schiaparelli  the 
weight  of  observational  evidence  favored  the  presence  of  this 
water  vapor.  Such  observational  evidence  is  all  obtained 
by  means  of  an  instrument  called  the  spectroscope  (p.  282). 
If  the  solar  light  reflected  from  Mars  passes  through  a  plane- 
tary atmosphere  containing  water  vapor,  its  "  spectrum," 
as  seen  in  the  spectroscope,  will  show  certain  bands  called 
water  vapor  bands.  Unfortunately,  we  cannot  observe 
the  Martian  light  until  it  has  passed  through  the  terrestrial 
atmosphere,  which  always  contains  some  water  vapor.  The 
difficulty  is  to  determine  whether  any  observed  vapor  bands 
are  due  to  the  Martian  atmosphere,  or  to  that  of  the  earth. 

There  is  but  one  way  to  distinguish  between  the  two  :  we 
must  compare  the  Martian  spectrum  with  that  of  our  moon. 
The  lunar  spectrum  will  show  no  water  vapor  effects  except 
such  as  are  due  to  the  earth's  air,  for  the  moon  itself  has  no 
atmosphere.  Consequently,  a  lunar  observation  gives  us 
only  terrestrial  water  vapor  bands ;  a  Mars  observation  gives 
us  the  terrestrial  plus  the  Martian  effects.  Any  observable 
difference  between  the  two  is  due  to  Martian  vapor  alone. 

It  is  clear  that  this  method  of  observation  cannot  be  suc- 
cessful if  there  is  very  much  water  vapor  in  our  own  air.  If 
there  is,  the  slight  difference  between  the  moon  and  Mars  will 
be  masked  completely.  As  existing  observations  have  been 
found  discordant,  Campbell  made  an  expedition  to  the 
summit  of  Mt.  Whitney  (15,000  ft.)  in  1909.  He  took  the 
necessary  instruments  with  him  and  photographed  the 
spectra  of  Mars  and  our  moon  at  the  exact  time  when  Mars 
was  most  favorably  situated  in  proximity  to  the  earth. 

224 


PLATE  8. 


Photos  by  Barnard. 

Mars  and  the  Crescent  Venus. 


THE   PLANETS   ONE   BY   ONE 

At  the  great  elevation  of  Mt.  Whitney  there  was  but 
little  terrestrial  atmosphere  above  the  observer.  The  lunar 
spectrum,  which  exhibits  the  effects  caused  by  our  own 
air,  showed  so  little  water  vapor  that  Campbell  concludes 
it  would  never  have  been  detected  at  all  by  a  person 
previously  ignorant  of  its  existence.  And  the  Martian 
spectrum  was  equally  destitute  of  water  vapor  bands, 
even  the  faintest.  On  account  of  this  elimination  of  the 
earth's  air,  these  observations  must  be  considered  by  far 
the  most  reliable  in  our  possession.  They  seem  to  settle 
the  water  vapor  question  in  the  negative ;  with  the  Martian 
vapor  the  Martian  water  goes;  and,  without  water,  the 
canals  are  impossible  as  artificial  waterways. 

The  markings  on  Mars  of  which  we  are  certain  consist  of 
various  permanent  patches,  lines,  and  areas  of  different 
shades;  and  there  are  also  two  bright  spots  at  the  poles 
known  certainly  since  the  time  of  Herschel.  The  accompany- 
ing Plate  8  contains  a  series  of  photographs  of  Mars,  showing 
these  markings  and  white  spots  very  plainly.  There  is  also 
distinct  evidence  of  axial  rotation,  the  exposures  having  been 
made  in  sets  of  three,  with  an  interval  of  lh  22m  between  the 
first  and  last  set.  In  that  interval  the  markings  have 
moved  perceptibly  across  the  disk  (p.  202).  The  lower 
part  of  the  plate  shows  a  photograph  of  Venus,  in  the 
crescent  phase  (p.  219).  The  polar  spots  seem  to  increase 
in  the  Martian  winter  season,  and  to  diminish  in  the  summer. 
If  so,  they  may  be  ice-caps ;  and  it  is  this  notion  that  gives 
color  to  the  canal  theory.  For  the  melting  ice-caps,  in 
this  theory,  are  used  as  the  source  of  water  to  be  pumped 
through  the  canals.  Later,  the  water  evaporated  out  of 
the  canals  is  supposed  to  be  returned  to  the  poles  by  atmos- 
pheric movements,  and  there  again  precipitated  as  snow. 

Q  225 


ASTRONOMY 

But  if  the  planet,  in  comparison  with  the  earth,  is  as  cold  as 
it  should  be  according  to  its  distance  from  the  sun,  there  is 
quite  a  possibility  that  the  caps  are  not  ice  at  all,  but  perhaps 
some  other  substance,  such  as  solidified  carbon  dioxide. 

It  is  not  difficult  to  calculate  the  theoretic  temperature  on 
the  surface  of  Mars,  and  it  is  found  to  be  —33°  on  the 
Fahrenheit  scale.1  Since  this  low  theoretic  temperature 
would  negative  the  existence  of  water  in  an  uncongealed 
state,  the  advocates  of  canals  are  compelled  to  assume 
a  heavy  blanket  of  atmosphere,  charged  with  much  water 
vapor,  to  keep  Mars  warm,  as  it  were,  and  cause  the  actual 
surface  temperature  to  be  far  above  its  theoretic  value. 
Such  an  atmosphere  might  act  in  this  way ;  but  here  again 
we  find  the  absolute  necessity  of  assuming  the  presence  of 
water  vapor  in  spite  of  observational  evidence  to  the  con- 
trary; and  we  are  asked  to  imagine  Mars  to  be  an  arid 
desert  requiring  irrigation,  and  yet  above  this  arid  desert 
a  wet,  foggy  atmosphere,  highly  charged  with  water  vapor. 

In  view  of  the  great  public  interest  in  this  Mars  matter, 
we  shall  venture  to  quote  briefly  from  an  article  published  by 
the  writer  a  few  years  ago.  First,  as  to  the  question :  Do 
the  geometric  markings  really  exist  ?  The  evidence  here  is 
almost  all  positive.  Most  astronomers  who  have  observed 
Mars  under  favorable  conditions  and  with  powerful  telescopes 
have  seen  markings,  but  the  number  of  lines  reported  varies 
from  several  hundreds  down  to  two  or  three.  Finally,  a 
very  few  prominent  markings  have  been  photographed. 

Let  us  first  consider  the  visual  evidence.  Let  us  examine 
the  witnesses,  for  that  is  what  these  astronomers  really  are, 
eye-witnesses  of  the  lines  on  Mars.  Some  years  ago, 
Lowell  observed  on  the  planets  Venus  and  Mercury  certain 

1  Note  31,  Appendix. 
226 


THE   PLANETS  ONE   BY  ONE 

systems  of  geometric  markings.  As  it  is  impossible  to 
suppose  that  all  planets  possess  intelligent  engineers,  it  is 
essential  to  the  Martian  theory  to  show  that  these  Venus 
markings  are  quite  unlike  those  now  seen  on  Mars.  Ac- 
cordingly, in  his  book  entitled  Mars  and  its  Canals,  pub- 
lished in  1906,  Lowell  refers  to  these  older  Venus  observa- 
tions in  the  following  words  :  ' i  The  Venusian  lines  are  hazy, 
ill-defined,  and  non-uniform."  But  in  the  original  article 
in  which  he  described  what  he  saw  on  Venus,1  we  find  the 
following  :  "They  (the  markings  on  Venus)  are  not  shadings 
more  or  less  definite,  but  perfectly  distinct  markings.  I 
have  seen  them  when  their  contours  had  the  look  of  a  steel 
engraving." 

The  only  way  in  which  these  two  statements  concerning 
Venus  can  be  brought  into  accord  is  to  suppose  that  in  the 
interval  of  nine  years  the  observer  has,  for  some  reason, 
changed  his  opinion. 

Additional  important  testimony  is  furnished  by  Mr.  A.  E. 
Douglass,  who  was  chief  assistant  at  the  Lowell  observatory 
in  Flagstaff  for  seven  years,  from  1894  to  1901,  and  since  then 
held  the  position  of  astronomer  there  for  a  considerable 
period.  In  May,  1907,  he  published  an  article  in  the  Popular 
Science  Monthly,  entitled  "Illusions  of  Vision,  and  the 
Canals  of  Mars."  This  title  alone  shows  that  he  had 
changed  his  views ;  and  his  actual  words  in  the  1907  article 
are : 

"The  ray  illusion  (sic)  is  to  me  a  very  satisfactory  explana- 
tion of  many  faint  canals  .  .  .  the  only  objective  reality 
is  the  spot  from  which  they  start." 

Again,  speaking  of  what  he  calls  the  "halo  illusion,"  he 
says : 

1  Monthly  Notices  of  the  Royal  Astronomical  Society,  London,  1897. 

227 


ASTRONOMY 

"The  double   canals  of  Schiaparelli  in   1881-2  and  of 
Perrotin  and  Thallon  in   1886  .      .  are  .      .  due   to   this 


cause/' 


And  again : 

"Thus,  in  conclusion,  we  see  that  there  are  fundamental 
defects  in  the  human  eye  producing  faint  canal  illusions." 

Having  thus  outlined  briefly  the  apparent  contradiction  in 
Lowell's  testimony  and  the  reversal  of  Douglass',  it  will 
be  of  interest  to  explain  how  it  may  be  possible  for  observers 
to  be  in  error  to  such  an  extent.  For  this  purpose,  we 
must  mention  some  of  the  possible  causes  that  may  impair 
the  correctness  of  an  observation.  At  least  five  im- 
perfections come  into  play :  imperfections  of  the  earth's 
atmosphere,  the  telescope,  the  eye,  the  optic  nerve,  and 
the  imagination.  The  process  of  seeing  a  thing  is  not 
at  all  simple.  Light  waves  coming  from  the  object  under 
examination,  after  passing  through  the  atmosphere  and 
telescope,  fall  upon  the  outer  surface  of  the  eye.  They  are 
concentrated,  or  focused,  by  the  lens  in  the  eye,  and  pro- 
duce an  effect,  which  we  do  not  quite  understand,  upon 
the  retina  at  the  back  of  the  eye.  This,  in  some  un- 
explained way,  results  in  an  impression  being  received 
by  the  brain  through  the  optic  nerve.  Then,  the  brain 
in  its  turn  does  an  unexplained  something  with  that  im- 
pression ;  what  we  think  we  see  is  equal  to  that  which 
came  through  the  eye  and  optic  nerve,  plus  what  the 
brain  does  with  it  later.  The  mind  cannot  distinguish 
between  an  impression  caused  by  the  eye  and  optic  nerve 
and  one  produced  by  action  of  the  brain  itself. 

Now  it  is  important  to  remember  that  imperfections  of 
the  atmosphere,  such  as  clouds,  and  all  imperfections  of 
the  telescope,  generally  tend  to  diminish  or  destroy  the 

228 


THE   PLANETS  ONE   BY  ONE 

possibility  of  vision ;  but  those  of  the  eye  and  imagination, 
if  they  act,  are  just  as  likely  to  increase  the  number  of 
details  we  think  we  see.  Especially  when  the  object  is  faint 
and  indistinct,  —  trembling,  as  it  were,  on  the  very  limit 
of  visibility,  —  then  especially  can  a  very  slight  activity 
of  the  imagination  either  prevent  our  seeing  it,  or  bring 
it  seemingly  into  view.  And  this  extreme  faintness  ad- 
mittedly exists  in  the  case  of  almost  all  the  Martian  mark- 
ings. 

This  theory  explains  why  highly  experienced  observers 
see  so  much  more  than  beginners.  They  think  they  are 
training  the  eye,  so  as  to  increase  its  powers,  while  in  reality 
they  may  only  be  training  that  slight  imperfection  of  the 
imagination  which  tends  to  increase  details  thought  to  be 
visible.  The  theory  also  furnishes  an  explanation  of  the 
fact  that  so  considerable  a  number  of  observers  think  they 
have  seen  the  faint  canals.  Nothing  more  strongly  increases 
the  powers  of  imaginary  seeing  —  of  seeing  the  unseen  — 
than  the  knowledge  that  others  have  already  made  the  ob- 
servation. We  are  very  prone  to  "see"  what  we  are  told 
by  others  is  visible  :  we  think  we  see  what  we  desire  and  hope 
to  see ;  do  what  we  will,  we  cannot  prevent  this. 

Coming  now  to  a  consideration  of  the  photographic  ob- 
servations, we  must  mention  one  or  two  matters  that  are 
not  well  known  to  the  general  public.  In  the  first  place 
the  size  of  a  Mars  picture  made  by  direct  exposure  of  a 
photographic  plate  at  the  focus  of  the  Lowell  telescopes  is 
not  larger  than  the  head  of  an  ordinary  pin.  From  so  small 
a  picture  we  could  not  even  hope  to  discover  any  details. 
Therefore  we  must  enlarge  it  as  much  as  possible ;  and  there 
are  two  ways  of  doing  this.  The  first  is  to  place  an  enlarging 
lens  in  the  telescope  itself.  Two  disadvantages  limit  this 

229 


ASTRONOMY 

method.  First,  it  complicates  the  optical  system  of  the 
telescope,  with  consequent  loss  of  distinctness  in  the  image  ; 
and  secondly,  it  makes  the  image  on  the  plate  less  brilliant. 
The  cause  of  this  loss  of  brilliancy  is  simple.  The  total 
quantity  of  light  received  from  the  planet  is  constant;  if, 
therefore,  we  spread  it  over  an  enlarged  surface,  each  part  of 
that  surface  will  receive  less  light.  For  instance,  with  an 
enlargement  of  five  diameters,  the  surface  of  the  image  is 
25  times  as  large.  The  resulting  diminution  of  light  makes 
necessary  a  longer  exposure  of  the  photograph,  and  a  con- 
sequent increased  difficulty  in  making  the  clock  mechanism 
attached  to  the  telescope  follow  with  exactness  the  motion  of 
Mars  in  the  sky.  Experiment  has  shown  the  greatest  photo- 
graphic enlargements  that  can  be  made  in  this  way  with 
the  Lowell  telescopes ;  and  the  negatives  of  Mars,  including 
the  most  recent  ones  made  in  the  Andes,  never  exceed  three- 
sixteenths  of  an  inch  in  diameter. 

The  other  method  of  increasing  the  size  of  photographs 
is  to  use  an  ordinary  enlarging  camera,  after  the  telescopic 
negative  has  been  finished.  There  is  here  no  difficulty  in 
securing  sufficient  light,  as  in  the  case  of  enlargements  made 
in  the  telescope  itself.  For  we  can  use  artificial  illumination 
of  the  original  negative,  and  make  this  illumination  as 
strong  as  may  be  necessary.  But  there  is  another  serious 
difficulty.  Every  photographic  negative  is  developed  by 
placing  the  plate  in  a  chemical  bath,  after  it  has  been  ex- 
posed to  light.  This  results  in  the  precipitation  of  silver 
particles  upon  the  plate,  wherever  its  sensitive  surface  has 
been  exposed  to  light.  The  picture  is  thus  built  up  of  sep- 
arate particles  of  silver.  These  particles  are  so  small 
that  the  eye  cannot  distinguish  the  separate  ones ;  they 
run  together,  as  it  were,  to  form  the  picture.  But  the  case 

230 


THE  PLANETS  ONE  BY  ONE 

is  very  different  if  we  magnify  the  negative.  We  then  see 
the  separate  grains  of  silver,  scattered  here  and  there  about 
the  surface,  and  the  picture  itself  is  lost  altogether.  The 
same  difficulty  occurs  if  we  attempt  to  examine  any  photo- 
graph with  an  ordinary  microscope  of  considerable  power. 
The  separate  silver  grains  at  once  appear,  and  the  picture 
effect  is  lost. 

All  this  photographic  experimentation,  therefore,  has  not 
yet  resulted  in  good  pictures  more  than  three-sixteenths 
of  an  inch  in  diameter  and  produced  by  purely  photo- 
graphic processes,  though  somewhat  larger  negatives  may 
possibly  be  made  in  the  future.1  All  larger  published 
pictures  have  been  reproduced  from  hand  drawings,  and 
are  therefore  simply  visual  observations.  The  alleged 
photographic  verifications  have  been  made  by  the  same 
observers  who  have  studied  Mars  in  the  visual  telescope; 
again  the  eye,  optic  nerve,  and  brain  were  brought  into  play, 
and  exactly  the  same  causes  as  before  impair  the  reliability 
of  these  visual  observations  of  photographs. 

We  conclude  that  neither  by  visual  nor  by  photographic 
evidence  has  the  existence  of  an  artificial  network  of  markings 
been  proven,  or  even  rendered  highly  probable.  Therefore 
the  time  has  not  yet  come  when  we  shall  have  to  inquire 
whether  geometric  lines  indicate  the  presence  of  intelligent 
inhabitants ;  that  time  will  arrive  if  the  lines,  themselves  are 
ever  shown  to  possess  a  real  or  even  a  highly  probable 
existence. 

We  shall  next  consider  the  Planetoids,  or  Minor  Planets 
(pp.  183,  196).  A  large  number  of  these  tiny  bodies  travel  in 

1  Those  shown  in  Plate  8,  p.  225,  made  with  the  40-inch  Yerkes'  tele- 
scope, are  about  twice  as  large  as  the  Lowell  photographs ;  and  they  show 
no  signs  of  geometric  canal  networks. 

231 


ASTRONOMY 

orbits  situated  between  Mars  and  Jupiter ;  up  to  the  present 
time  several  hundreds  have  been  discovered  and  their  orbits 
and  motions  computed. 

Much  interest  attaches  to  the  history  of  the  first  one 
ever  observed,  —  Ceres.  It  had  long  been  noted  that  the 
space  between  Mars  and  Jupiter  was  an  exceptionally 
large  empty  space  in  the  solar  system ;  and  it  seemed  strange 
that  no  planet  should  exist  there.  The  matter  appeared 
still  more  peculiar  after  Bode's  empirical  law  was  published 
(p.  196)  in  the  latter  part  of  the  eighteenth  century ;  for  this 
law  indicated  that  there  should  be  a  planet  between  Mars 
and  Jupiter.  And  in  1781  this  indication  received  stronger 
confirmation,  when  the  older  Herschel  found  Uranus,  one 
of  the  modern  exterior  planets,  in  or  near  the  position 
predicted  by  the  law.  An  astronomical  society  was  accord- 
ingly organized  to  make  a  systematic  search  for  the  expected 
unknown  planet.  But  not  until  the  first  day  of  the  nine- 
teenth century  did  the  long-sought  object  reveal  itself; 
and  to  an  independent  observer  in  the  Italian  city  of  Palermo. 
There  Piazzi  was  making  an  accurate  catalogue  of  the  fixed 
stars.  Every  night  he  made  telescopic  observations  from 
which  he  could  compute  stellar  right-ascensions  and  dec- 
linations (p.  34) ;  and  he  planned  to  enter  in  a  catalogue 
these  two  coordinates  for  every  star  in  the  sky,  bright  enough 
to  come  within  the  range  of  his  telescopic  power. 

But  he  did  not  confine  himself  to  single  observations. 
Each  night's  work  was  checked  by  careful  repetition  on 
several  other  nights.  Sometimes  he  found  an  error,  which 
usually  consisted  in  the  discovery  of  a  star  that  had  escaped 
his  notice  at  some  previous  observation.  But  on  this  historic 
occasion,  he  found  that  a  star  was  absent,  although  he  had 
observed  it  on  another  night.  And,  strange  to  say,  there 

232 


THE   PLANETS  ONE   BY  ONE 

was  also  an  additional  star  close  by,  one  that  had  apparently 
remained  unobserved  on  the  previous  night.  The  conclusion 
was  irresistible  that  the  new  star  was  the  same  one  he  had 
observed  before,  and  that  it  must  have  moved  among  the 
other  stars  in  the  interval.  This  motion  among  the  stars 
(p.  10)  is  the  distinguishing  characteristic  of  planets.  A 
third  observation  made  the  matter  sure :  the  second  star 
was  again  absent,  and  a  third  new  star  once  more  appeared 
in  a  place  previously  vacant.  The  apparent  motion  between 
the  second  and  third  observations  was  proportional,  both  in 
magnitude  and  direction,  to  that  between  the  first  and  second 
observations.  So  it  must  surely  be  a  wandering  star,  —  a 
true  planet.  Discovery  and  fame  were  his. 

But  Piazzi  was  able  to  observe  the  new  planet  during  a 
few  weeks  only,  on  account  of  illness.  When  news  reached 
the  astronomers  of  northern  Europe,  Ceres  had  already 
passed  so  near  conjunction  (p.  209)  with  the  sun  that  further 
observations  were  impossible.  There  was  well-grounded 
fear  that  the  planet  would  not  be  found  again ;  for  astrono- 
mers at  that  time  had  no  good  method  of  determining  a 
planet's  orbit  from  observations  extending  through  such  a 
short  time.  The  older  planets  had  been  observed  through 
many  complete  revolutions,  and  there  was  never  any  danger 
of  their  being  lost,  because  they  are  bright  enough  to  be 
visible  easily  by  the  unaided  eye. 

But  there  was  a  young  astronomer  at  Gottingen,  Gauss  by 
name,  who  succeeded  in  solving  this  difficult  problem ; 
and  from  his  published  orbit  and  ephemeris  it  was  easy 
to  find  the  planet  again  as  soon  as  the  apparent  motion  of 
the  sun  in  the  ecliptic  had  brought  the  planet  to  a  position 
where  it  could  again  be  sought  in  darkness. 

A  year  later,  in  1802,  the  second  minor  planet  Pallas  was 

233 


ASTRONOMY 

found.  In  1804  Juno  was  added ;  and  in  1807,  Vesta.  It 
was  not  until  1845  that  another  appeared ;  and  three  more 
in  1847.  From  that  time  on  discovery  proceeded  but 
slowly,  because  the  method  in  use  was  still  the  tiresome  and 
arduous  process  employed  by  Piazzi.  But  in  1891,  Wolf 
attacked  the  problem  photographically,  the  photographic 
method  having  just  commenced  to  be  widely  applied  to 
astronomic  purposes.  His  procedure  was  perfectly  simple. 
A  photographic  plate  was  exposed  in  the  telescope  for 
several  hours ;  and  care  was  taken  to  make  sure  that  clock- 
work attached  to  the  telescope  moved  it  accurately  during 
those  hours,  so  as  to  keep  pace  exactly  with  the  diurnal 
rotation  of  the  celestial  sphere. 

The  photograph,  when  developed,  would  of  course  show  a 
round  dot  corresponding  to  each  fixed  star  within  the  field 
of  view  of  the  telescope.  But  if  there  was  a  wandering  planet 
in  range,  it  would  move  slightly  with  respect  to  the  stars 
during  the  period  of  photographic  exposure ;  and  conse- 
quently its  image  in  the  picture  would  be  drawn  out  into  a 
short  line,  instead  of  appearing  as  a  round  dot  like  the 
stellar  images.  Thus  the  presence  of  a  line  would  infallibly 
betray  the  existence  of  a  planet.  As  many  as  seven  plane- 
toids have  been  thus  found  on  a  single  plate ;  so  the  method 
is  enormously  effective.  To  it  we  owe  an  immense  increase  in 
our  minor  planet  knowledge  during  the  past  twenty  years. 

Plate  9  is  a  photographed  field  of  stars,  with  two  planetoid 
lines,  or  "  trails. "  They  will  be  found  near  the  middle  of 
the  picture,  as  indicated  by  the  marginal  arrows.  The 
trails  are  not  quite  parallel,  showing  that  the  orbit  planes  of 
these  two  planetoids  are  inclined  at  slightly  different  angles  to 
the  ecliptic  plane.  The  difference  in  thickness  of  the  trails 
indicates  a  difference  of  luminosity  in  the  two  planetoids. 

234 


I 


Photo  by  Barnard. 


PLATE  9.     Discovery  of  Planetoids. 


THE   PLANETS  ONE   BY  ONE 

The  orbits  of  the  small  planets  present  some  interesting 
peculiarities.  There  are  several  open  spaces  where  practi- 
cally no  orbits  appear.  Curiously  enough,  these  open 
spaces  occur  at  points  where  the  minor  planet  periods  of 
orbital  revolution  in  accordance  with  Kepler's  harmonic  law 
(p.  188)  bear  a  simple  relation  to  the  period  of  Jupiter. 
It  was  long  ago  explained  by  Lagrange  that  if  two  planets 
have  periods  connected  by  a  simple  proportion,  such  as  f,  3, 
|,  etc.,  then  persistent  perturbations  (p.  206)  must  result, 
which  will  gradually  change  the  orbits  until  the  simple 
relation  is  destroyed.  It  is  in  accord  with  this  principle 
that  Jupiter  has  forced  the  minor  planet  orbits  out  of  these 
critical  positions  in  space,  and  made  them  congregate  at 
intermediate  positions. 

As  to  the  size  of  the  planetoids,  it  has  been  computed 
that  the  mass  of  the  entire  group  can  be  but  a  small  fraction 
of  the  earth's  mass.  The  individual  planetoids  are  probably 
not  more  than  one  ten-thousandth  as  massive  as  the  earth, 
and  their  diameter  will  not  average  more  than  twenty  miles. 

As  to  the  evolution  of  these  minor  planets,  there  is  not  much 
doubt.  If  we  accept  the  hypothesis  of  Laplace,  usually 
known  as  the  Nebular  Hypothesis,  the  planets  were  formed 
by  the  concentration  of  matter  thrown  off  from  the  sun  in 
early  ages,  while  it  was  still  in  a  gaseous  or  nebulous  con- 
dition. This  matter  is  supposed  to  have  been  detached 
from  the  central  mass  in  the  form  of  a  ring :  we  have  only 
to  imagine  the  minor  planets  an  exceptional  case,  in  which 
the  ring,  after  breaking  up,  was  prevented  from  concentrating 
into  a  single  body  by  the  perturbative  action  of  the  big  planet 
Jupiter.  Any  other  hypothesis  as  to  the  early  history  of 
the  planets  must,  of  course,  also  explain  the  planetoids  as 
an  exceptional  case  in  cosmic  evolution. 

235 


ASTRONOMY 

Among  the  minor  planets  is  one  very  remarkable  one, 
discovered  by  Witt  in  1898  and  by  him  named  Eros.  Its 
orbit  comes  well  within  that  of  Mars,  and  it  approaches  the 
earth  at  times  nearer  than  any  other  planetary  body  except 
our  own  moon.  It  can  pass  within  about  13 \  million  miles 
of  the  earth ;  and  this  makes  it  an  especially  valuable  planet  to 
observe  for  the  purpose  of  ascertaining  by  certain  indirect 
methods  the  distance  from  the  earth  to  the  sun.  It  is 
altogether  probable  that  observations  of  Eros  will  give  us 
ultimately  the  most  accurate  value  of  the  sun's  distance 
yet  attained.  There  will  be  a  very  favorable  opportunity 
to  attempt  the  necessary  observations  in  1931  (cf.  p.  263). 

Proceeding  outward  from  the  sun,  we  now  reach  the 
planet  Jupiter,  the  largest  in  the  solar  system,  and  the  most 
brilliant  object  in  the  sky  at  night,  with  the  possible  occa- 
sional exception  of  Venus.  In  the  telescope  Jupiter  is  a 
magnificent  object,  second  only  to  Saturn  in  interest.  It 
surpasses  Saturn  in  size,  but  it  lacks  the  splendid,  calm 
mysterious  ring.  Markings  of  a  more  or  less  permanent 
character  exist ;  they  look  like  cloud-belts  running  along  the 
planet's  equator.  And  clouds  they  doubtless  are ;  for  Jupiter 
must  have  retained,  and  must  possess,  a  deep  layer  of  atmos- 
phere, on  account  of  his  very  high  gravitational  attraction 
(see  pp.  167,  222) ;  and  since  there  is  also  a  high  albedo,  or 
reflecting  power,  we  should  expect  the  outer  surface  to  be 
made  of  clouds,  which  have  this  power  in  a  high  degree. 

Jupiter's  rotation  period,  or  day,  can  of  course  be  deter- 
mined from  the  markings.  It  is  9h  55m ;  but  there  is  some 
uncertainty  in  this  period,  because  the  cloud-markings 
probably  have  a  drift  of  their  own  on  the  planet's  surface, 
and  thus  do  not  determine  the  rotation  with  precision. 
The  axis  is  only  3°  out  of  perpendicularity  with  the  orbit 

236 


THE   PLANETS  ONE   BY  ONE 

plane ;  so  that  there  should  be  no  considerable  seasonal 
differences  of  temperature.  But  the  average  Jovian  tempera- 
ture must  be  very  high,  for  the  constant  visibility  of  clouds 
indicates  a  hot  surface  temperature.  If  this  be  correct, 
Jupiter  must  be  in  a  condition  slightly  resembling  that  of 
the  sun.  It  must  furnish  its  own  heat,  for  it  is  too  far 
from  the  sun  to  receive  much  thermal  assistance  from  that 
body. 

Jupiter  has  eight  moons,  of  which  four  can  be  seen  with 
a  small  telescope.  They  are  very  interesting  historically, 
for  their  discovery  in  1610  by  Galileo  gave  its  death-blow 
to  the  old  Ptolemaic  theory  of  the  universe.  The  follow- 
ing is  a  brief  account  of  this  great  discovery,  partly  quoted 
from  an  article  by  the  writer,  first  printed  in  the  New  York 
Evening  Post. 

What  must  have  been  Galileo's  feelings  when  he  first 
found  with  his  "new"  telescope  the  satellites  of  Jupiter? 
They  were  seen  on  the  night  of  Jan.  7,  1610.  He  had  al- 
ready viewed  the  planet  through  his  earlier  and  less  powerful 
glass,  and  was  aware  that  it  possessed  a  round  disk  like  the 
moon,  only  smaller.  Now  he  saw  also  three  objects  that  he 
took  to  be  little  stars  near  the  planet.  But  on  the  following 
night,  the  three  small  stars  had  changed  their  positions,  and 
were  now  all  situated  to  the  west  of  Jupiter,  whereas  on  the 
previous  night  two  had  been  on  the  eastern  side.  He  could 
not  explain  this  phenomenon,  but  he  recognized  that  there 
was  something  peculiar  at  work.  Long  afterwards,  in  one 
of  his  later  works,  translated  into  quaint  old  English  by 
Salusbury,  he  declared  that  "one  sole  experiment  sufficeth 
to  batter  to  the  ground  a  thousand  probable  Arguments." 

The  9th  was  cloudy,  but  on  the  10th  he  again  saw  his 
little  stars,  their,  number  now  reduced  to  two.  He  guessed 

237 


ASTRONOMY 

that  the  third  was  behind  the  planet's  disk.  The  positions 
of  the  two  visible  ones  were  altogether  different  from  either 
of  the  previous  observations.  On  the  llth  he  became  sure 
that  what  he  saw  was  really  a  series  of  satellites  accom- 
panying Jupiter  on  his  journey  through  space,  and  at  the 
same  time  revolving  around  him.  In  the  12th,  at  3  A.M.,  he 
actually  saw  one  of  the  small  objects  emerge  from  behind 
the  planet ;  on  the  13th  he  finally  saw  four  satellites.  Two 
hundred  and  eighty-two  years  were  destined  to  pass  away 
before  any  human  eye  should  see  a  fifth.  It  was  Barnard,  in 
1892,  who  followed  Galileo. 

To  understand  the  effect  of  this  discovery  upon  Galileo 
requires  a  person  who  has  himself  watched  the  stars,  not  as 
a  dilettante,  seeking  recreation  or  amusement,  but  with 
that  deep  reverence  that  comes  only  to  him  who  feels  —  nay, 
knows  —  that  in  the  moment  of  observation  just  passed 
he,  too,  has  added  his  mite  to  the  great  fund  of  human  knowl- 
edge. Galileo  knew,  on  that  llth  of  January,  1610,  that 
the  memory  of  him  would  never  fade ;  that  the  very  music 
of  the  spheres  would  thenceforward  be  attuned  to  a  truer 
note,  if  any  would  but  hearken  to  the  Jovian  harmony. 
For  he  recognized  that  the  visible  revolution  of  these  moons 
around  Jupiter,  while  that  planet  was  itself  visibly  traveling 
through  space,  must  end  the  old  Ptolemaic  theory  of  the 
universe.  Here  was  a  great  planet,  the  center  of  a  system 
of  satellites,  and  yet  not  the  center  of  the  universe.  Surely, 
then,  the  earth,  too,  might  be  a  mere  planet  like  Jupiter,  and 
not  the  supposed  motionless  center  of  all  things. 

The  most  interesting  phenomena  about  the  Jovian  satel- 
lites are  their  frequent  eclipses  and  transits  (p.  13).  Any 
satellite  may  be  eclipsed  to  us,  either  through  passing 
behind  the  ball  of  the  planet,  or  by  moving  into  such  a 

238 


THE   PLANETS  ONE  BY  ONE 

position  that  the  planet  interposes  between  the  satellite 
and  the  sun.  In  the  latter  case,  the  satellite  receives  no 
sunlight  to  reflect  in  our  direction,  and  so  becomes  invisible. 

At  other  times  a  satellite  will  " transit"  between  us  and 
Jupiter.  Then  it  generally  becomes  invisible,  too,  unless 
it  happens  to  be  projected  against  a  dark  part  of  the  Jovian 
surface,  such  as  one  of  the  cloud-belts.  Finally,  a  satellite 
may  pass  between  Jupiter  and  the  sun,  when  its  shadow  is 
thrown  on  the  planet's  surface,  and  is  plainly  visible  as  a 
round  black  dot  slowly  crossing  the  bright  planetary  disk. 
None  of  the  transits  or  eclipses  occur  suddenly :  the  satellites 
all  have  disks  of  sensible  magnitude,  and  thus  encroach  upon 
the  planet's  edge  very  gradually. 

Observations  of  the  exact  time  of  these  satellite  eclipses 
are  useful  as  an  easy  approximate  method  of  determining 
terrestrial  longitudes.  If  we  note  the  instant  of  local  time 
when  an  eclipse  occurs,  and  compare  it  with  the  calculated 
Greenwich  time,  as  given  in  the  astronomical  almanac,  we 
ascertain  at  once  the  time  difference  of  the  observer's 
position  on  the  earth,  measured  from  Greenwich.  And  this, 
multiplied  by  15,  gives  his  longitude  at  once  in  degrees 
(p.  74).  This  method  requires  no  instruments  beyond  an 
ordinary  small  telescope;  but  it  is  not  very  precise  on  ac- 
count of  the  impossibility  of  observing  the  exact  instant 
when  the  eclipse  happens.  Somewhat  higher  precision, 
with  equal  simplicity  in  method,  may  be  secured  from  ob- 
servations of  star  occultations  (p.  166).. 

It  was  from  observations  of  Jupiter's  satellite  eclipses  that 
Roemer,  in  1675,  first  ascertained  that  light  is  not  propagated 
through  space  instantaneously,  but  requires  an  appreciable 
time  for  its  transmission.  He  used  a  long  series  of  satellite 
eclipses,  and  found  they  did  not  succeed  each  other  at  equal 

239 


ASTRONOMY 


intervals.  During  half  the  year  they  came  too  soon,  by 
gradually  increasing  amounts;  during  the  other  half-year 
they  came  too  late,  by  similar  quantities  of  time.  Roemer 
soon  found  that  they  came  too  soon  when  the  earth  was 
approaching  Jupiter ;  too  late  when  it  was  increasing  its  dis- 
tance from  Jupiter.  He  concluded  correctly  that  they  came 
too  soon  because  the  earth's  approach  to  Jupiter  diminished 
the  distance  the  light  traveled  before  reaching  the  earth,  after 
the  eclipse  actually  occurred.  Consequently,  the  light 

arrived  at  the  earth 
sooner,  and  observers 
saw  the  event  sooner, 
too.  It  is  clear  from 
Fig.  62  that  the  extreme 
difference  between  ac- 
celerated and  retarded 
eclipses  must  be  the 
quantity  of  time  required 
by  light  to  cross  a  diame- 
ter EiEz  of  the  earth's  or- 
bit. This  is  found,  by  ob- 
servation of  the  eclipses, 
to  be  998  seconds.  Itfol- 
an  eclipse  will  be  observed 
998  seconds  sooner  than  it  would  have  been  observed  if  the 
earth  had  remained  at  Ez  instead  of  traveling  around  its 
orbit.  Bradley  kn^w  of  Roemer 's  observations  when  he 
explained  aberration  (p.  136),  nearly  a  century  later,  as  a 
result  depending  on  the  velocity  of  light  and  the  earth's 
velocity  in  its  orbit. 

Since  modern  laboratory  experiments  have  made  known 
that  light  moves  at  the  rate  of  186,000  miles  per  second,  we 

240 


FIG.  62.    Roemer's  Discovery. 


lows  that  when  the  earth  is  at 


THE  PLANETS  ONE   BY  ONE 

have  only  to  multiply  this  number  by  998  to  obtain  the  diam- 
eter of  the  earth's  orbit  in  miles.  This  gives  186,000,000 
miles,  approximately ;  and  this  number  is  correct. 

The  planet  Saturn  is  the  last  of  the  large  planets  observed 
by  the  ancients.  The  most  interesting  thing  about  it  is  its 
magnificent  ring  system, —  a  series  of  three  disk-like  rings, 
situated  nearly  in  the  plane  of  the  planet's  equator.  The 
history  of  their  discovery  is  worth  noting.  We  hear  of  them 
first  from  Galileo,  who  saw  a  couple  of  " handles"  or  ansce 
attached  to  the  planet  in  1610.  He  was  unable  to  explain 
them ;  and,  when  he  looked  for  them  again  on  a  later  date, 
was  unable  to  see  them  at  all.  The  story  is  that  he  gave 
them  up  as  inexplicable. 

Nearly  half  a  century  later,  in  1656,  Huygens  published  a 
book  De  Saturni  Luna  Observatio  Nova,  in  which  he  announced 
the  discovery  of  a  satellite,  and  also  gave  a  correct  explana- 
tion of  the  mysterious  ansce.  But  Huygens  was  not  quite 
certain  that  his  explanation  was  right.  He  was  most 
anxious  to  secure  for  himself  the  priority  of  discovery,  and 
yet  he  was  unwilling  to  make  a  premature  and  possibly  in- 
correct announcement.  So  he  resorted  to  the  ingenious 
device  of  a  "logogriph,"  or  puzzle.  It  appears  in  the 
De  Saturni  Luna  as  follows : 1 

aaaaaaa        ccccc         d         eeeee         g        h        iiiiiii 
1111    mm        nnnnnnnnn         oooo       pp      q    rr      s 
ttttt  uuuuu 

It  was  not  until  1659,  three  years  later,  in  a  book  entitled 
Systema  Saturnium,  that  Huygens  rearranged  the  above 
letters  in  their  proper  order,  reading : 2 

1  It  may  be  found  in  's  Gravesande's  ^dition  of  Huygens,  Lugduni  Ba- 
tavorum,  1751,  p.  526. 

2  Same  edition,  p.  566. 

B  241 


ASTRONOMY 

"  Annulo  cingitur,  tenui  piano,  nusquam  cohaerente,  ad 
eclipticam  inclinato.' ' 

At  the  same  time/  he  re-published  a  series  of  drawings 
exhibiting  several  incorrect  interpretations  of  the  ring 
phenomena,  as  observed  by  various  older  astronomers. 
These  are  reproduced  in  Plate  10 :  Fig.  1  is  by  Galileo,  ob- 
served 1610 ;  Fig.  2,  by  Scheiner,  1614 ;  Figs.  3,  8,  9,  13, 
by  Ricciolus,  1640-1650;  Figs.  4,  5,  6,  7  by  Hevelius; 
Fig.  10,  by  "Eustachius  de  Divinis,"  1646-1648 ;  Fig.  11,  by 
Fontana,  1646 ;  Fig.  12,  by  Gassendi  and  Blancanus.  Under 
these  reproductions  from  Huygens  we  have  placed  a  fine 
drawing  made  by  Barnard  with  the  Yerkes  40-inch  telescope, 
Dec.  12,  1907. 

It  will  be  observed  that  by  the  publication  of  the  logo- 
griph  of  1656,  Huygens  secured  for  himself  the  credit  of 
what  he  had  done.  If  any  other  astronomer  had  pub- 
lished the  true  explanation  after  1656,  Huygens  could  have 
proved  his  claim  to  priority  by  re-arranging  the  letters  of  his 
puzzle.  On  the  other  hand,  if  further  researches  showed  that 
his  explanation  was  wrong,  he  would  never  have  made 
known  the  true  meaning  of  his  logogriph,  and  would  thus 
have  escaped  the  ignominy  due  to  publishing  an  erroneous 
explanation.  So  the  method  of  announcement  was  com- 
parable in  ingenuity  with  the  Huygenian  explanation  itself. 

The  ring  phases  admit  of  easy  explanation.  The  rotation 
axes  of  all  revolving  bodies  maintain  constant  directions  in 
space,  unless  disturbed  by  attractions  such  as  cause  the 
earth's  axis  to  produce  the  precession  of  the  equinoxes  (p. 
129).  Therefore  the  plane  of  the  rotating  rings  must  like- 
wise always  maintain  an  unvarying  direction  in  space.  Now 
if  this  plane  of  the  rings  is  imagined  extended  outward, 

1  Same  edition,  p.  634. 
242 


PLATE  10.     Saturn. 


THE  PLANETS  ONE  BY  ONE 

until  it  cuts  the  celestial  sphere,  it  will  trace  out  a  great 
circle  there.  This  circle  necessarily  meets  the  ecliptic 
circle  in  two  opposite  points  (cf.  Fig.  6,  p.  35),  which  are 
called  nodes;  and  it  so  happens  that  the  angle  between 
the  great  ring  circle  and  the  ecliptic  is  28°  on  the  celestial 
sphere. 

Saturn  revolves  in  its  orbit  around  the  sun  in  a  period  of 
about  30  years.  Therefore,  it  must  pass  one  of  the  nodes 
every  fifteen  years,  approximately.  When  Saturn  is  thus 


FIG.  63.    Phases  of  Saturn's  Ring. 


projected  at  one  of  the  nodes,  the  sun,  in  its  apparent 
motion  along  the  ecliptic,  may  happen  to  appear  in  the 
other  node  at  the  same  time.  These  positions  are  illustrated 
in  Fig.  63  ;  but  in  using  this  figure,  it  must  not  be  forgotten 
that  Saturn's  orbit  around  the  sun  is  very  nearly  in  the 
ecliptic  plane,  in  which  the  earth's  orbit  is  also  located.  Let 
the  sun,  then,  be  at  $,  and  the  earth  at  E.  Thus  the  sun  ap- 
pears projected  on  the  celestial  sphere  at  the  node  £'. 
Saturn,  located  at  H  in  its  orbit,  is  projected  at  the  same  time 
in  the  other  node  at  H' '.  It  is  evident  that  the  earth  must 
then  lie  on  the  line  HS',  the  intersection  of  the  two  planes, 

243 


ASTRONOMY 

so  that  it  is  temporarily  in  the  plane  of  the  ring  as  well  as 
in  the  ecliptic  plane. 

When  the  earth  is  thus  in  the  ring  plane,  we  must  see  the 
ring  edgewise ;  and  it  is  so  thin  that  it  then  becomes  quite 
invisible,  except  to  the  most  powerful  modern  telescopes. 
It  disappears,  as  Galileo  found.  Furthermore,  near  these 
times  of  disappearance,  the  earth  may  be  for  a  short  time 
on  either  side  of  the  ring  plane.  And  unless  Saturn  is  quite 
accurately  at  the  node,  the  sun  will  also  be  a  little  on  one  side 
of  the  ring  plane  or  on  the  other.  But  the  ring  is  illuminated 
on  one  side  only, —  the  side  toward  the  sun.  Consequently,  if 
the  sun  happens  to  be  on  one  side  of  the  ring  plane  while  the 
earth  is  on  the  other,  we  observe  the  dark  side  of  the  ring- 
system.  It  should  then  also  be  invisible ;  but  powerful 
telescopes  will  still  show  it,  appearing  like  a  fine  line  of 
light.  This  is  well  seen  in  Barnard's  drawing  (Plate  10, 
p.  242),  together  with  certain  condensations,  or  thick  places 
in  the  ring.  This  drawing  was  made  with  the  ring  in  the 
edgewise  phase  :  in  Plate  11,  we  have  added  a  fine  photo- 
graph, also  by  Barnard,  showing  the  open  phase.  This 
negative  was  made  Nov.  19,  1911,  with  the  60-inch  reflecting 
telescope  at  the  Mt.  Wilson  observatory  in  California. 

As  we  have  found,  these  times  of  ring-disappearance  occur 
about  once  every  fifteen  years.  In  years  near  the  periods 
of  disappearance,  the  ring  is  seen  nearly  edgewise :  it  then 
looks  like  a  very  narrow  oval  or  ellipse ;  and  it  opens  out 
to  the  widest  extent  about  seven  or  eight  years  on  either 
side  of  the  date  of  disappearance.  But  the  ring  can  never 
open  into  a  circle,  for  the  earth  can  never  be  elevated  more 
than  28°  above  the  plane  of  the  ring,  since  28°  is  the  angle 
between  the  ring  plane  and  the  ecliptic  plane,  in  which  the 
earth  is  always  situated.  And  the  earth  would  need  to  be 

244 


THE   PLANETS   ONE   BY  ONE 

elevated  90°  above  the  ring  plane  to  enable  us  to  see  the 
ring  as  a  circle. 

As  to  the  constitution  of  the  rings,  we  have  very  certain 
and  most  interesting  knowledge.  They  are  neither  solid, 
liquid,  nor  gaseous,  but  consist  of  a  dense  swarm  of  tiny 
satellites,  moving  in  orbits  closely  interwoven,  and  all 
lying  in,  or  near,  the  plane  of  Saturn's  equator.  They  are 
so  numerous,  and  their  orbits  so  closely  packed  and  in- 
tertwined, that  we  cannot  see  between  them,  and  so  they 
look  like  a  solid  disk.  They  are  not  very  unlike  the  group 
of  planetoids,  which  are  known  to  encircle  the  sun  be- 
tween the  orbits  of  Mars  and  Jupiter  (p.  231).  In  1857, 
Clerk-Maxwell  proved  mathematically  that  it  is  impossible 
for  a  system  of  solid  or  liquid  rings  to  exist  permanently. 
They  would  be  in  unstable  equilibrium,  and  must  infallibly 
break  into  a  series  of  satellites.  And  this  mathematical 
demonstration  was  abundantly  verified  observationally  in 
1896,  by  Keeler.  He  observed  the  ring  on  both  sides  of  the 
planet  with  the  spectroscope.  With  this  instrument,  to  be 
described  later,  it  is  possible  to  measure  the  linear  velocity 
with  which  the  edges  of  the  ring  approach  the  earth,  or  re- 
cede from  it,  as  the  ring  performs  its  axial  rotation  around 
the  polar  axis  of  Saturn.  Now  it  can  be  shown  mathemati- 
cally that  if  the  ring  is  really  a  mass  of  satellites,  its  outside 
edge  should  rotate  more  slowly  than  its  inside  edge.1  On  the 
other  hand,  if  the  ring  is  solid,  of  course  the  outside  must  move 
faster  than  the  inside.  Keeler  found  by  actual  measurement 
that  the  outside  of  the  ring  was  moving  10  miles  per  second  ; 
the  inside,  12J  miles ;  and  he  thus  verified  observationally 
the  correctness  of  Clerk-Maxwell's  mathematical  conclusion. 

This  observation  of  Keeler's  is  destined  to  rank  as  a 

1  Note  32,  Appendix. 
245 


ASTRONOMY 

classic  observation.  We  are  given  to  regard  astronomy  as 
an  ancient  science,  long  since  perfected,  and  incapable  of 
further  progress  of  importance.  But  this  analysis  of  the 
ring  constitution  by  methods  purely  mathematical;  this 
theoretic  prediction  of  invisible  relative  velocities  of  rotation ; 
and,  finally,  the  complete  observational  verification  by  a 
method  essentially  novel,  —  all  this  constitutes  a  chain  of 
scientific  research  worthy  of  standing  at  the  side  of  the 
master  work  of  the  seventeenth  century. 

Saturn  has  ten  moons,  in  addition  to  the  swarm  composing 
the  ring  system.  The  largest  (discovered  by  Huygens)  is 
visible  in  small  telescopes.  Five  were  found  before  1700; 
Herschel  found  two  in  1789 ;  Bond  one  in  1848 ;  the  other 
two  were  discovered  photographically  at  Harvard  College 
observatory  within  recent  years. 

The  next  planet,  Uranus,  was  discovered  by  Sir  William 
Herschel  in  1781.  The  histqry  of  Herschel,  and  of  this  dis- 
covery, is  not  without  interest.  He  was  the  son  of  a  Ger- 
man musician,  was  born  in  1738,  and  came  to  England  in  1757 
to  seek  his  fortune.  He  settled  at  Bath,  where  he  supported 
himself  successfully  as  a  music  teacher.  Although  he  worked 
very  hard  at  his  music,  he  found  time  to  study  also  his 
favorite  sciences  of  mathematics  and  astronomy.  Having 
no  instrument,  he  decided  to  make  one ;  but  it  was  not  until 
1774  that  he  succeeded  in  constructing  a  tolerable  reflecting 
telescope.  He  wrote  in  1783:  "I  determined  to  accept 
nothing  on  faith,  but  to  see  with  my  own  eyes  what  others 
had  seen  before  me."  Four  times  he  made  a  new  telescope, 
each  of  greater  size  than  the  last ;  and  with  each  he  made  a 
re-survey  of  the  entire  visible  heavens.  On  his  tomb  is 
graven  the  epitaph : 

"  Coelorum  perrupit  claustra." 
246 


THE   PLANETS  ONE  BY  ONE 

It  was  in  the  second  of  his  celestial  reviews,  made  with 
an  instrument  only  seven  feet  long,  that,  as  he  says,  "in 
examining  the  small  stars  in  Gemini,  I  perceived  one 
that  appeared  visibly  larger  than  the  rest.  I  suspected 
it  to  be  a  comet."  Within  a  short  time,  Lexell  was  able 
to  show  that  the  motions  of  the  new  object  could  not 
be  explained  by  any  cometary  orbit,  and  that  it  must 
be  a  new  planet. 

It  was  perhaps  the  most  startling  discovery  ever  made  in 
astronomy  :  Herschel  named  it  the  Georgium  Sidus,  in  com- 
pliment to  the  English  king,  who  promptly  honored  him 
with  an  appointment  at  court,  and  made  him  rich  with  a 
pension  of  £200.  He  removed  to  Slough,  near  Windsor, 
where  he  built  " Observatory  House,"  and  made  it  memor- 
able as  the  scene  of  endless  important  astronomical  dis- 
coveries. Long  afterwards,  Arago  characterized  it  as  "le 
lieu  du  monde  ou  il  a  etefait  le  plus  de  decouverles." 

Uranus  has  four  satellites,  two  discovered  by  the  same 
Herschel  in  1787,  and  two  by  Lassell  in  1851.  They  have 
one  important  peculiarity :  they  revolve  in  their  orbits 
around  the  planet  from  east  to  west  instead  of  west  to  east, 
the  usual  direction  of  orbital  motion  in  the  solar  system. 
They  are  thus  an  exceptional  case,  and  constitute  in  a  way 
an  unexplained  difficulty  in  the  Laplacian  nebular  hypothe- 
sis (p.  235),  which  would  seem  to  require  all  satellites  to 
revolve  in  the  same  direction. 

The  outermost  known  planet  is  Neptune,  remarkable 
principally  on  account  of  the  interest  attaching  to  its  dis- 
covery. Shortly  after  Uranus  had  been  found,  astronomers 
searched  their  old  records,  and  ascertained  that  good  ob- 
servations of  it  existed  as  early  as  1690.  But  it  had  always 
passed  for  a  star,  its  disk  not  being  big  enough  to  betray  its 

247 


ASTRONOMY 

planetary  character  on  sight,  even  in  the  telescope.  But  no 
orbit  could  be  found  which  would  bring  these  early  observa- 
tions into  accord  with  the  numerous  ones  which  began  to  be 
accumulated  immediately  after  discovery.  And  the  planet 
soon  refused  to  live  up  to  its  modern  observations,  also. 
More  than  one  astronomer  suggested  that  there  must  be  an 
unknown  planet  exterior  to  Uranus,  and  perturbing  its  mo- 
tion, so  as  to  throw  it  alternately  in  advance  of  its  proper 
orbital  position  and  behind  it. 

In  1845,  a  young  Englishman,  Adams,  who  had  graduated 
from  Cambridge  University  only  two  years  before,  succeeded 
in  constructing  an  orbit  for  the  hypothetical  exterior  planet, 
basing  his  calculations  simply  on  the  observed  discrepancies 
in  the  orbital  motion  of  Uranus.  He  wrote  to  the  astronomer 
royal  at  Greenwich,  asking  him  to  look  for  the  new  object 
with  his  big  telescope  in  a  certain  definite  position  on  the  sky.' 
We  now  know  that  this  position  given  by  Adams  was  correct 
within  2°,  so  that  a  little  careful  " sweeping"  with  the  tele- 
scope would  undoubtedly  have  revealed  the  planet.  But  the 
astronomer  royal  made  an  unfortunate  mistake  ;  the  story  is 
that  he  delayed  attending  to  Adams'  letter. 

But  another  astronomer,  Leverrier,  was  also  working 
at  the  problem  in  France ;  by  August,  1846,  he,  too,  had 
worked  out  the  new  orbit.  On  the  23d  of  September  a 
letter  from  him  arrived  in  Berlin  and  was  delivered  to 
Galle  at  the  observatory  there.  Galle  had  a  new  and  very 
complete  star-chart  of  the  proper  region  of  the  sky ;  and  it 
was  for  this  reason  that  Leverrier  had  written  to  him,  rather 
than  to  any  other  astronomer.  As  soon  as  it  became  dark, 
Galle  went  into  the  observatory  dome,  and  began  to  compare 
his  chart  with  the  sky.  He  very  soon  found  a  strange  body 
within  less  than  1°  of  the  exact  spot  indicated  in  Leverrier's 

248 


THE   PLANETS   ONE  BY  ONE 

letter.  It  was  an  exciting  moment;  the  new  planet  had 
been  seen  at  last. 

One  curious  fact  is  that  both  Adams  and  Leverrier  made 
use  of  Bode's  law  (p.  196)  to  obtain  an  approximate  value 
for  the  supposed  planet's  distance  from  the  sun.  This  law 
has  no  foundation  in  theory ;  but  it  had  proved  to  be  fairly 
exact  for  all  planets  then  known,  including  Uranus.  But 
it  fails  for  Neptune ;  and  accordingly  both  the  computers  were 
very  largely  in  error  as  to  this  important  element  of  their  new 
planet's  orbit.  There  is  ground  for  supposing  that  their 
success  was  due  in  some  degree  to  accidental  favoring 
circumstances.  But  the  result  was  unquestionably  a  great 
triumph  for  mathematical  science  and  for  Newton's  law  of 
gravitation :  at  this  distance  of  time  it  is  proper  to  divide 
the  honor  of  the  discovery  equally  between  Adams  and 
Leverrier.  It  is  certainly  great  enough  for  two  men,  but 
in  the  middle  of  the  last  century,  an  ascerbitous  controversy 
raged  about  the  assignment  of  priority  in  this  matter. 

Neptune  is  so  far  away  from  the  earth  that  but  few  de- 
tails have  been  discovered  concerning  it.  There  is  but  one 
known  satellite.  And  beyond  Neptune,  no  further  planets 
have  been  found,  though  the  existence  of  such  "ultra-Nep- 
tunian" bodies  has  often  been  suspected.  But  none  has  ever 
been  revealed,  even  to  the  most  careful  photographic  surveys 
so  far  made  in  the  heavens. 

But  there  is  one  other  material  substance  in  the  solar 
system  that  requires  mention  here.  The  mysterious  Zodiacal 
Light  is  observable  as  a  faintly  luminous  band  traceable 
along  the  ecliptic  circle  outward  from  the  sun  for  a  consid- 
erable distance  both  east  and  west.  There  is  also  at  times 
a  faint  glow  called  by  the  German  name  "  Gegenschein " 
discernible  in  the  part  of  the  ecliptic  opposite  the  sun.  The 

249 


ASTRONOMY 

whole  thing  may  perhaps  be  best  explained  as  a  ring  of 
excessively  minute  planets,  revolving  around  the  sun  in 
an  orbit  larger  than  that  of  the  earth.  Those  near  the  sun 
would,  of  course,  be  the  brightest ;  and  the  Gegenschein  would 
be  the  combined  effect  of  an  infinitude  of  these  particles 
acting  like  tiny  full-moons  in  their  position  of  opposition 
(p.  163)  to  the  sun. 


250 


CHAPTER  XIII 

THE    TIDES 

KEPLER  was  probably  the  first  man  to  notice  that  the 
tides  of  ocean  are  due  to  some  form  of  attraction  exerted 
by  the  moon.  He  looked  upon  the  moon  as  a  personal  ally 
of  the  earth,  and  in  his  quaint  Latin  remarks  that  "a  mutual 
affection  between  allied  bodies  tends  towards  their  union. " 
It  is  possible  that  Kepler  may  have  had  some  hazy  idea  of 
gravitation  as  a  species  of  personal  characteristic  of  celestial 
bodies. 

Let  us  begin  by  summarizing  the  facts  easily  observable 
by  any  one  who  examines  the  behavior  of  the  ocean  along 
its  shores.  In  the  first  place,  it  will  be  found  that  the  water 
level  changes  considerably.  During  six  hours,  approxi- 
mately, the  waters  rise ;  and  again,  for  about  six  hours,  they 
fall.  In  each  day  there  are  ordinarily  two  high  tides  and  two 
low  tides.  Furthermore,  in  addition  to  merely  rising  and 
falling,  the  water  also  flows  along  the  coast,  in  one  direc- 
tion during  a  period  equal  to  the  time  of  rising  tide,  more 
or  less,  and  in  the  opposite  direction  during  a  period  corre- 
sponding in  duration  to  the  falling  tide.  Thus  strong  tidal 
currents  exist ;  and  navigators  frequently  take  advantage 
of  them  to  increase  the  speed  of  ships,  especially  in  the  case 
of  sailing  vessels  engaged  in  the  difficult  business  of  beating 
(as  it  is  called)  against  an  adverse  wind. 

We  shall  consider  the  earth  for  a  moment  as  a  globe 
uniformly  covered  with  a  shallow  ocean.  The  most  im- 

251 


ASTRONOMY 

portant  cause  of  tides  on  such  an  imaginary  globe  would 
be  the  gravitational  attraction  of  the  moon.  We  know, 
from  Newton's  law,  that  such  gravitational  attraction  dimin- 
ishes rapidly  with  an  increase  in  the  distance  separating  the 
attracted  particle  from  the  moon.  Consequently,  the  moon 

attracts  the  water  on  the  earth's 
surface  more  strongly  than  it 
does  the  more  distant  solid  earth 
beneath  it.  We  should  there- 
fore expect  the  moon  to  heap 

The  Tides.  UP    Watei>    at    that    P°mt    On    the 

earth   which   is   nearest  to   the 

moon.  But  there  is  also  water  on  the  earth  on  the  side 
opposite  the  moon ;  and  this  water  is  attracted  less  than  the 
solid  earth ;  it  is  attracted  least  of  any  terrestrial  material, 
because  it  is  farthest  of  all  from  the  moon.  In  other  words 
(Fig.  64),  the  moon  should  pull  the  water  away  from  the 
earth  at  M,  tending  to  heap  it  up ;  and  at  0,  it  should  pull 
the  earth  away  from  the  water,  again  tending  to  heap  it  up. 
But  the  tidal  forces  exerted  by  the  moon  do  not  act  in 
the  above  very  simple  way ;  in  fact,  the  actual  heaping  up  of 
the  water  due  to  the  above  cause  would  be  quite  insignificant. 
A  far  greater  effect  is  produced  at  points  not  directly  under 
the  moon.  Here  the  tidal  force  is  not  vertical,  because  the 
moon  is  not  directly  overhead  ;  it  may  be  regarded  as  divided 
between  the  vertical  and  horizontal  directions.  And  the 
horizontal  fraction  is  then  usually  the  important  one.  It 
tends  to  move  particles  of  water  horizontally  along  the 
earth's  surface,  and  to  move  them  toward  the  place  where  the 
moon  is  overhead.  But  owing  to  the  earth's  axial  rotation, 
the  moon  rises  in  the  east  and  sets  in  the  west.  While  it  is 
east  of  the  meridian  in  any  given  place,  it  is  pulling  the 

252 


THE  TIDES 

water  particles  eastward  with  the  horizontal  fraction  of 
the  force.  After  crossing  the  meridian,  it  of  course  pulls 
them  westward  ;  and  therefore  the  result  should  be  an  oscilla- 
tion of  the  water  particles  backward  and  forward,  occupy- 
ing approximately  half  a  day  to  go  and  come.  And  of  course 
this  means  half  a  lunar  day  (p.  176) ;  not  twelve  ordinary 
solar  hours. 

Since  the  moon,  at  any  given  moment,  is  east  of  certain 
places  on  the  earth,  and  west  of  other  places,  it  follows  that 
different  parts  of  the  ocean  will  be  oscillating  different  ways 
at  the  same  time.  This  must  produce  a  tidal  wave,  with 
high  tide  at  the  place  where  the  crest  of  the  wave  is  situated. 
The  crest  would  follow  the  moon  around,  as  the  earth  ro- 
tates, but  it  would  not  be  under  the  moon.  It  can  be  shown 
that  it  would,  in  fact,  ordinarily  be  90°  distant  from  the 
point  under  the  moon.  And  there  would  be  a  second  crest 
opposite  the  first,  according  to  reasoning  similar  to  that  of 
Fig.  64 ;  and  therefore  two  daily  high  tides,  following  the 
moon  around. 

Having  thus  outlined  the  explanation  of  the  semi-diurnal 
tide,  of  which  the  period  is  half  the  lunar  day,  it  is  now 
possible  to  explain  that  the  two  tides  each  day  are  of  un- 
equal size.  In  general,  one  rises 
higher  than  the  other.  Although 
we  have  seen  that  the  high  tidal 
crest  is  not  under  the  moon,  we  can 
still  reason  as  if  it  were,  in  order  to 
explain  the  above  inequality.  As 
the  earth  rotates  on  an  axis  perpen-  FlG-  65-  Dmmai  inequality 

of  Tides. 

dicular  to  the  plane  of  the  terrestrial 

equator,  every  point  on  the  earth's  surface  must  rotate  in  a 
circle  parallel  to  the  equator.  Now,  in  Fig.  65,  supposing 

253 


ASTRONOMY 

the  tidal  crests  to  be  under  the  moon  and  opposite  it,  and 
the  earth  rotating  around  the  axis  NS,  a  person  at  P  will  not 
have  as  high  a  high  tide  as  he  will  have  twelve  hours  later, 
when  the  earth's  rotation  has  carried  him  around  to  P'. 
The  difference  is  shown  by  the  dotted  lines  at  P  and  P',  and 
is  called  the  diurnal  inequality  of  the  tides.  It  is  due,  of 
course,  to  the  moon's  not  being  situated  in  the  plane  of  the 
equator.  If  the  moon  were  in  the  equator  plane,  the  tidal 
crests  would  be  placed  symmetrically  with  respect  to  the 
equator,  and  the  two  high  tides  would  be  practically  equal. 
In  fact,  it  must  happen  on  two  days  each  month  that  the 
moon  really  is  in  the  equator  plane.  For  the  moon's  ap- 
parent motion  on  the  sky,  due  to  its  orbital  motion  around 
the  earth,  appears  to  take  place  in  a  great  circle  of  the  celes- 
tial sphere  (p.  160),  which  must,  of  course,  cut  the  celestial 
equator  at  two  points.  And  it  is  a  fact  in  accord  with 
actual  tidal  observation  that  the  diurnal  inequality  disap- 
pears twice  each  month.  Then  the  two  tides  are  equal. 

To  complete  this  part  of  the  subject,  two  more  details 
must  be  mentioned.  First,  we  recall  that  the  lunar  orbit 
around  our  earth  is  an  oval  or  ellipse,  with  the  earth 
at  one  focus,  at  some  distance  from  the  center.  The 
moon  will  therefore  be  especially  near  the  earth  at  certain 
times.  When  it  is  at  the  nearest  point  of  its  orbit,  the 
perigee  (p.  169),  its  tide-raising  force  will  be  greater  than  at 
any  other  time.  In  fact,  this  force  is  about  |  greater  at 
perigee  than  at  apogee.  Perigee  and  apogee,  of  course,  both 
occur  each  month ;  consequently,  the  high  tides  are  by  no 
means  equally  high  during  the  entire  month. 

The  other  matter  requiring  mention  is  the  effect  of  the 
sun  on  the  tides.  It  operates  in  a  manner  precisely  similar 
to  the  moon,  but  its  greater  distance  diminishes  the  tidal 

254 


THE  TIDES 

force,  so  that  the  solar  tide  is  only  about  T\  of  the  lunar  tide. 
The  sun  is  far  larger  than  the  moon,  but  its  greater  gravi- 
tational attraction  due  to  mass  or  bulk  is  more  than  counter- 
balanced by  its  greater  distance.  But  it  is  clear  that  when 
the  sun  and  moon  are  so  placed  that  they  act  together,  we 
shall  get  especially  high  tides ;  and  when  they  act  against 
each  other,  the  tides  will  be  especially  feeble. 

Of  course  these  two  bodies  pull  together  when  sun,  earth, 
and  moon  are  situated  more  or  less  in  a  single  straight  line ; 
and  this  occurs  at  the  dates  of  full  and  new  moon  (p.  163). 
We  then  have  the  great  tides  called  Spring  Tides ;  and  when 
the  moon  is  in  the  first  and  third  quarters  we  have  the  little 
tides  called  Neap  Tides.  Spring  and  neap  tides  have  rela- 
tive heights  in  the  ratio  of  11  +  5  to  11  —  5,  or  8  to  3,  be- 
cause the  solar  pull  is  T\  of  the  lunar. 

The  above  brief  outline  of  tidal  theory  is  greatly  modi- 
fied when  applied  to  the  actual  earth,  upon  which  the 
oceans  are  deep  bowls,  large,  but  still  limited  in  size; 
and  the  gulfs,  sounds,  etc.,  small,  shallow  limited  cups.  The 
laws  of  wave  motion  in  areas  limited  as  to  size  and  depth 
come  into  play :  according  to  these  laws,  the  rate  of  progress 
of  a  wave,  or  its  time  of  oscillation,  depends  on  the  depth 
of  water.  For  instance,  in  a  basin  like  the  north  Atlantic 
Ocean,  a  wave  would  move  500  miles  per  hour  if  it  were 
set  in  motion,  and  then  left  to  itself.  It  would  pass  from 
Europe  to  America  in  about  six  hours.  Thus  its  period  for 
going  and  returning  would  be  nearly  equal  to  the  tidal 
period  of  half  a  lunar  day.  It  can  be  demonstrated  that 
when  these  two  periods  are  thus  about  equal,  the  tides  will 
be  large.  The  water  will  practically  oscillate  about  a 
neutral  line  in  the  middle  of  the  ocean,  giving  high  tide  at 
the  European  coast  when  it  is  low  tide  at  the  American. 

255 


ASTRONOMY 

But  this  explanation  is  complicated  still  further  by  the 
configuration  of  the  coasts,  whereby  the  Atlantic  does  not 
act  as  a  single  basin  with  a  single  neutral  line,  but  as  several 
basins  overlapping,  more  or  less.  But  the  general  result  is 
nearly  as  stated. 

When  we  come  to  the  peculiar  tides  belonging  to  limited 
areas  of  the  coast,  —  such  a  basin  as  Long  Island  Sound,  for 
instance,  —  still  a  different  explanation  is  required.  Here  the 
tidal  wave  is  not  really  due  directly  to  the  moon ;  it  is  a  special 
local  oscillation,  set  up  by  contact  or  impact  from  the 
lunar  tide  in  the  ocean  outside  the  sound.  In  such  cases, 
conditions  become  quite  complicated,  and  often  lead  to 
tides  much  higher  than  the  ocean  tides.  For  instance,  in  Long 
Island  Sound  the  rise  and  fall  is  about  seven  feet ;  high  tide 
occurs  at  nearly  the  same  time  throughout  the  sound ;  and 
the  wave  motion  produces  a  rapid  current,  or  motion  of  the 
water  particles  along  the  sound.  Another  interesting  tidal 
modification  is  found  in  the  funnel-shaped  Bay  of  Fundy, 
where  the  tides  rise  and  fall  as  much  as  forty  feet,  or  even 
more. 

Tidal  phenomena  produce  results  of  importance  other  than 
recurrent  changes  in  the  oceans.  Tidal  evolution  is  a  term 
used  to  describe  effects  produced  on  the  earth  as  a  whole  by 
tidal  action  continued  throughout  vast  ages  of  time.  It 
is  clear  that  the  tidal  motions  of  great  masses  of  water  must 
consume  a  vast  quantity  of  mechanical  energy,  especially 
where  the  great  tidal  currents  occur  along  the  ocean  coasts. 
In  such  cases  there  must  also  be  much  friction  between  the 
land  and  water.  Friction  will  generate  heat,  and  consume 
more  energy. 

Now  all  this  energy  must  be  derived  from  some  source : 
the  law  of  the  conservation  of  energy  (p.  2)  tells  us  that  there 

256 


THE  TIDES 

can  be  no  manifestation  of  new  energy,  such  as  we  have 
just  mentioned,  without  an  equal  and  corresponding  dim- 
inution of  the  manifestation  of  energy  somewhere  else. 
The  place  where  we  should  expect  to  find  this  diminution 
is  in  the  earth's  rotation.  In  other  words,  we  should  expect 
tidal  friction,  etc.,  to  act  as  a  sort  of  brake  on  the  earth's 
axial  rotation,  and  to  bring  about  a  consequent  minute 
lengthening  of  the  terrestrial  day,  after  the  lapse  of  sufficient 
ages  of  time.  But  the  most  delicate  astronomic  observations 
have  failed  to  detect  any  such  lengthening  of  the  day.  It 
must  therefore  be  extremely  small,  certainly  not  more  than 
TOFO  o"  °f  a  second  in  a  century. 

But  there  is  another  interesting  consequence  of  these  con- 
siderations :  how  does  terrestrial  tidal  friction  affect  the 
moon's  motions  ?  In  Fig.  66,  the  moon  is  shown  in  the 
celestial  equator,  and  the 
two  great  tidal  protuber- 
ances at  H  and  Hf.  Ac- 
cording to  theory,  as  we 
have  seen  (p.  253),  these 
protuberances  or  tidal 
crests  should  be  at  A  and 
B,  90°  from  the  point 

.  FIG.  66.    Tidal  Effect  on  the  Moon. 

under    the    moon.     But 

so  much  of  the  tidal  effect  as  acts  like  friction,  to  retard 
the  terrestrial  rotation,  must  also  make  the  two  protuber- 
ances lag  behind  their  proper  positions  at  A  and  B.  This 
brings  H  nearer  the  moon  than  H' ;  increases  the  lunar 
attraction  at  H,  as  compared  with  H' ;  and  therefore  ac- 
celerates the  moon's  motion  in  its  orbit,  as  shown  by  the 
arrow. 

Now   it  can  be   demonstrated    from   the   principles    of 

s  257 


ASTRONOMY 

mechanics,  that  increasing  the  velocity  of  a  body  moving  in 
an  orbit  will  increase  also  the  size  of  the  orbit  and  the  period 
of  revolution  of  the  body  in  the  orbit.  Therefore,  tidal 
friction  must  make  the  moon  recede  from  the  earth,  and 
must  also  lengthen  the  lunar  sidereal  period.  In  the  hands 
of  G.  H.  Darwin,  these  simple  principles  have  led  to  an 
extremely  plausible  theory  as  to  the  formation  of  our  moon. 
According  to  Darwin,  the  moon  once  formed  part  of  the 
earth ;  the  entire  mass  was  in  a  semi-liquid  or  plastic  con- 
dition ;  and  was  in  quite  rapid  rotation  about  an  axis.  There 
was  a  tremendous  flattening  of  the  earth  at  the  poles,  due 
to  plasticity.  It  can  be  shown  mathematically  that  such  a 
rotatory  flattened  plastic  body  may  assume  any  one  of 
several  shapes.  One  of  these  possible  figures  is  pear-shaped. 
The  fact  that  we  have  a  moon  is  thought  by  Darwin  to 
prove  that  the  pear-shaped  figure  actually  was  the  one  that 
happened  to  prevail.  The  rotating  pear-shaped  figure  should 
then  pass  over  into  an  hour-glass ;  from  that  to  a  dumb-bell, 
with  unequal  weights  at  the  ends.  Finally  comes  a  separa- 
tion; a  true  planet  with  a  moon,  both  revolving  rapidly 
about  their  common  center  of  gravity,  and  very  near  each 
other. 

Now  come  gigantic  tides ;  tides  compared  with  which  our 
present  ocean  tides  are  absolutely  insignificant.  For  the 
plastic  earth  was  subject  to  great  bodily  tides,  not  merely 
little  oscillations  of  a  thin  shell  of  ocean.  Frictional  forces 
then  produced  no  mere  slight  perturbative  action  •  they  were 
dominating  forces.  The  moon  was  driven  farther  and 
farther  from  the  earth ;  and  the  lunar  sidereal  period  was 
lengthened,  until  both  bodies  reached  the  condition  now 
existing. 

If  this  theory  is  correct,  it  enables  us  to  predict  for  future 

258 


THE   TIDES 

ages  the  final  condition  of  our  moon,  when  the  last  stage  of 
equilibrium  shall  finally  prevail.  When  that  occurs,  the 
lunar  sidereal  period  and  the  terrestrial  day  will  be  equal, 
the  earth  rotating  on  its  axis  in  55  of  our  present  days,  and 
the  moon  making  an  orbital  revolution  around  the  earth 
in  precisely  the  same  period.  The  moon  should  then  be 
always  opposite  the  same  point  of  the  earth's  surface;  and 
both  bodies  should  revolve  as  though  both  were  attached 
rigidly  to  the  ends  of  an  unbending  bar. 


259 


CHAPTER  XIV 

THE    SOLAR    PARALLAX 

WE  have  had  occasion  to  mention  several  times  the  im- 
portance of  a  correct  knowledge  of  the  distance  separating 
the  earth  from  the  sun.  In  our  discussions  of  planet &ry 
motions  we  have  considered  this  distance  to  be  known ; 
in  fact,  we  have  assumed  all  the  elements  (p.  200)  of  the 
earth's  orbit  to  be  within  our  knowledge. 

Until  the  latter  part  of  the  eighteenth  century,  astronomers 
had  only  a  very  rough  knowledge  of  the  sun's  distance,  or 
of  its  Parallax.  This  last  term  may  be  defined  easily ;  it  is 
exactly  analogous  to  the  corresponding  term  already  defined 
(p.  169)  in  the  case  of  the  moon.  By  the  solar  parallax  we 

simply  mean  half 
the  angular  diame- 
ter (p.  118)  of  the 

Fzo.67.    Solar  Parallax.  6arth>     ^Upposed     to 

be    seen    from    the 

sun.  Thus,  in  Fig.  67,  imagine  an  observer  situated  on  the 
sun  at  S.  Draw  a  straight  line  from  S  to  the  center  of  the 
earth  at  C  and  another  to  the  surface  of  the  earth  at  0. 
Then  the  angle  at  S  between  these  two  lines  is  half  the  angu- 
lar diameter  of  the  earth  as  seen  from  the  sun,  and  is  there- 
fore the  solar  parallax.  A  simple  equation  exists,1  by  means 

1  If,  in  Fig.  67,  we  let  D  represent  the  sun's  distance;  TT,  the  parallax 
angle ;  and  r  the  earth's  radius  ;  we  have  at  once,  from  the  triangle  SCO  : 

tan  TT  ^  -£y  or  D  =  ^^'     (Cf.  Note  20,  Appendix.) 

260 


THE   SOLAR   PARALLAX 

of  which  we  can  calculate  the  solar  parallax  from  the  solar 
distance,  or  the  distance  from  the  parallax.  If  either  be 
known,  we  can  at  once  find  the  other.  So  the  term  "  solar 
parallax"  is  really  a  substitute  for  " solar  distance."  The 
two  terms  are  interchangeable,  in  a  way;  but  they  are 
not  synonymous.  One  is  an  angle,  the  other  a  linear  dis- 
tance. The  present  accepted  value  of  the  solar  parallax  is 
8."80. 

We  shall  now  consider  various  ways  of  measuring  it. 
The  reader  will  remember  the  method  already  mentioned 
(p.  168)  for  ascertaining  the  moon's  distance  by  simultaneous 
observations  of  that  body  from  two  observatories,  one  in  a 
high  northern  latitude  on  the  earth,  and  the  other  in  a  high 
southern  latitude.  Of  course  this  same  method  might  be 
applied  to  the  sun,  but  there  is  an  objection  that  renders  it 
almost  useless.  This  objection  arises  from  the  small  size 
of  the  parallax  angle,  which  is  really  the  quantity  to  be 
measured.  The  lunar  parallax  is  about  1°,  the  solar  only 
8.  "8 ;  consequently,  any  small  error  of  observation,  such  as 
one-tenth  of  a  second  of  arc,  will  have  a  considerable  effect 
in  the  case  of  the  solar  parallax,  while  it  would  be  quite  in- 
appreciable in  the  case  of  the  moon. 

This  difficulty  can  be  obviated  in  some  degree  by  measur- 
ing the  solar  distance  in  an  indirect  manner.  The  distance 
is  really  only  used  to  ascertain  the  scale,  or  size,  of  the 
planetary  orbits.  For  with  the  aid  of  Kepler's  harmonic 
law  (p.  188),  we  can  find  the  relative  distances  of  the  various 
planets  from  the  sun,  after  we  have  observed  their  periods. 
Then,  knowing  the  distances,  we  can  make  a  map  of  all  the 
orbits,  here  once  more  supposed  to  be  concentric  circles. 
And,  again  with  our  knowledge  of  the  periods,  we  can  locate 
the  planets  themselves  in  their  orbits  on  the  map,  for  any 

261 


ASTRONOMY 

date,  if  we  have  also  observed  the  dates  of  conjunctions, 
etc.  (p.  209).  Such  a  map  will  be  correct  in  every  respect, 
except  that  the  scale  remains  unknown.  The  heavenly 
bodies  will  be  shown  in  their  proper  relative  places  on  the 
date  in  question,  but  we  do  not  know  the  number  of  miles 
corresponding  to  an  inch  on  the  map ;  in  other  words,  the 
scale  of  the  map.  To  ascertain  this,  it  will  be  sufficient  to 
measure  observationally  the  distance  from  the  earth  to  any 
other  planet  on  the  date  for  which  the  map  was  drawn.  This 
distance  once  known  from  the  observation  in  miles,  every 
other  distance  on  our  map  of  the  solar  system  also  becomes 
known  in  miles. 

This  work  will  be  most  accurate,  if  we  select  for  measure- 
ment a  planet  which  comes  comparatively  near  the  earth, 

and  make  our  observations 
and  our  map  at  the  time  of 
closest  approach.  For, 
after  all,  distance  from  the 
earth  can  be  measured  only 
by  using  the  earth's  own 
diameter  in  the  way  sur- 
veyors  use  what  they  call  a 
"base-line."  Thus,  in  Fig. 

68,  for  a  planet  at  P,  we  can  at  best  only  measure  the  angles 
POM  and  PMO,  so  as  to  ascertain  the  planet's  distance  by 
constructing  the  triangle  POM  from  the  known  base  OM  (the 
earth's  diameter)  and  the  two  measured  angles.1  But  the 
base  OM  is  always  necessarily  wofully  short,  compared  with 
the  planet's  distance;  the  slightest  errors  in  the  measured 
angles  produce  very  large  errors  in  the  distance.  Therefore 

1  We  can,  of  course,  substitute  a  solution  of  the  triangle  by  trigonometry 
for  the  geometrical  construction. 

262 


THE   SOLAR  PARALLAX 

we  must  do  this  work  when  we  can  observe  a  planet  that 
is  as  near  to  us  as  planets  are  ever  found. 

The  planet  Mars  has  been  used  with  advantage.  .  A 
time  is  selected  when  Mars  is  in  opposition  (p.  212)  so  that  it 
comes  to  the  meridian  at  midnight,  and  can  therefore  be 
observed  almost  all  night.  And  an  opposition 
is  chosen,  too,  when  the  earth  has  one  of  its 
closest  approaches  to  Mars.  This  combina- 
tion of  conditions  gives  the  most  favorable 
state  of  affairs  for  the  desired  measurements.  F  69  Favor 
Figure  69  shows  the  positions  of  the  sun,  earth,  able  Opposition 
and  Mars  at  the  time  of  opposition.  The 
orbits  are  not  circles,  and  therefore  the  distances  of  the  two 
planets  from  the  sun  are  variable.  If  the  opposition  is  one 
at  which  the  earth  happens  to  be  at  its  greatest  distance 
from  the  sun  (aphelion),  and  Mars  at  its  closest  possible 
approach  to  the  sun  (perihelion,  p.  120),  the  distance  between 
the  earth  and  Mars  is  as  small  as  it  can  ever  be ;  and  the  con- 
ditions are  especially  favorable  for  its  precise  measurement. 

Having  thus  secured  a  favorable  opposition,  there  are  two 
different  ways  in  which  the  Martian  distance  can  be  observed. 
We  may  employ  a  modification  of  the  method  already 
described  for  the  moon  (p.  168),  and  observe  Mars  from 
two  terrestrial  observatories  situated  as  far  apart  as  possible. 
In  that  case  our  base-line  is  the  line  joining  the  two  ob- 
servatories. Or  we  can  use  the  "diurnal"  method.  In  this 
method,  the  planet  is  observed  from  the  same  place  on  the 
earth  at  two  different  times  on  the  same  night :  first,  shortly 
after  sunset ;  and  second,  shortly  before  sunrise.  In  the 
interval,  the  rotation  of  the  earth  on  its  axis  will  carry  the 
observer  to  a  different  position  in  space ;  and  the  line  joining 
his  two  positions  becomes  the  base-line. 

263 


ASTRONOMY 


Thus,  in  Fig.  70,  let  us  disregard  the  slow  orbital  motions 
of  Mars  and  the  earth,  since  these  will  amount  to  but  little 
in  the  few  hours  elapsing  between  the  two  observations ; 
and  consider  the  earth's  diurnal  rotation  only.  Let  the 
earth's  center  be  at  (7,  with  the  rotation  axis  passing  through 


FIG.  70.    Diurnal  Method. 

C  perpendicular  to  the  printed  page;  and  the  observer  at 
Ow.  Then  the  observer  will  be  carried  in  about  ten  hours 
from  Ow  to  Oe  by  the  diurnal  rotation,  and  the  length  of  the 
line  OwOe  can  be  calculated  easily  from  the  known  dimen- 
sions of  the  earth,  and  the  time  elapsing  between  the  two 
observations.  This  line  OwOe  becomes  the  base-line.  From 
the  point  Ow  we  see  the  planet  projected  on  the  celestial 
sphere  in  the  direction  MW ;  from  Oe  we  see  it  in  the  direc- 
tion ME.  The  difference  of  the  two  measured  directions  is 
the  angle  OwMOe ;  from  this,  together  with  the  known  base- 
line OwOe,  we  can  calculate  the  distance  MC  from  Mars  to 
the  earth. 

In  making  the  observations,  both  for  the  diurnal  method 
and  for  the  method  with  two  observatories,  the  most  accurate 
way  to  observe  is  to  use  as  an  auxiliary  some 
small  star  appearing  near  Mars  on  the  sky. 
In  Fig.-  71,  which  represents  a  part  of 
the  sky,  such  a  star  is  shown.     In  either 
method  of   observation,   owing  to  the   ob- 
server's change  of  position  from  one  end  of 
the  base-line  to  the  other,  the  observations  show  the  planet 

264 


Observa- 
tion of  Mars. 


THE   SOLAR  PARALLAX 

projected  at  two  slightly  different  positions  on  the  sky,  as 
MI  and  M%.  The  star  itself  is  always  seen  in  the  same  posi- 
tion, because  the  stars  are  all  practically  infinitely  distant 
in  comparison  with  any  base-line  available  on  the  earth. 

The  angular  distance  on  the  sky  between  the  star  and  Mars 
(or  its  equivalent,  the  difference  in  direction  of  the  star  and 
Mars  as  seen  from  the  earth)  can  be  measured  in  the  same 
way  that  the  angular  diameter  of  a  planet  is  measured 
(p.  203),  with  an  instrument  called  a  Micrometer,  to  be 
described  later. 

Thus  we  obtain  from  the  two  observations  the  angular 
distances  of  MI  and  M2  from  the  star.  Their  difference  is 
the  arc  M iMz  on  the  sky ;  and  this  is  the  angle  OeMOw  of 
Fig.  70,  or  the  angle  subtended  by  the  base-line  OeOw  at 
the  distance  of  Mars  from  the  earth.  Comparatively  simple 
calculations  will  then  transform  this  angle  into  a  knowledge 
of  the  Martian  distance,  the  Martian  parallax,  and  thence 
the  solar  parallax. 

The  diurnal  and  the  two-observatory  methods  each  have 
advantages  and  disadvantages.  In  the  diurnal  method,  the 
observations  can  all  be  made  by  one  man  in  one  place  with  one 
instrument.  This  eliminates  those  errors  that  arise  from  per- 
sonality of  the  observer,  and  differences  between  different  in- 
struments, etc.  On  the  other  hand,  the  two  observations  are 
necessarily  separated  by  several  hours  in  time,  while  they  can 
be  made  quite  simultaneous  in  the  two-observatory  method. 
In  our  present  discussion,  we  have  neglected  totally  the  slight 
orbital  motions  of  Mars  and  the  earth.  This  is  without 
effect  if  the  two  observations  are  simultaneous ;  but  they 
never  are  so  in  the  diurnal  method.  The  two  planetary 
motions  must  be  taken  into  account  by  calculation ;  and 
thus  any  slight  existing  errors  in  our  supposed  knowledge  of 

265 


ASTRONOMY 

the  two  planetary  orbits  produce  slight  indirect  inaccuracies 
in  the  resulting  parallax  determination. 

But  in  the  two-observatory  method  it  is  by  no  means 
easy  to  secure  perfectly  simultaneous  observations,  either. 
The  two  stations  are  very  far  apart  on  the  earth.  The 
weather  is  quite  likely  to  be  cloudy  at  one  station  when  it 
is  clear  at  the  other,  thus  preventing  simultaneous  results. 
Indeed,  vagaries  of  weather  sometimes  seem  especially 
designed  to  hinder  astronomers  in  their  work,  particularly 
when  simultaneous  observations  are  required.  But  in  the 
diurnal  method,  the  astronomer  carries  his  weather  with  him, 
as  it  were,  while  he  is  rotated  by  the  earth  from  one  end 
of  his  base-line  to  the  other.  If  he  begins  with  a  good  clear 
night,  he  is  quite  likely  to  secure  the  necessary  corresponding 
second  observation. 

The  best  measurements  of  Mars  by  the  diurnal  method 
were  made  by  Gill  in  1877.  He  organized  a  special  astro- 
nomical expedition  to  the  island  of  Ascension  in  the  south 
Atlantic ;  and  this  was  an  especially  favorable  spot  for  his 
purpose.  It  was  desirable  to  be  near  the  equator,  so  that 
the  diurnal  base-line  might  be  a  long  one.  For  at  the  pole, 
of  course,  the  diurnal  circle  shrinks  into  a  mere  point. 

Gill  obtained  the  value  8/78  for  the  sun's  parallax  from 
the  Ascension  expedition,  but  it  appeared  that  various 
causes  interfered  to  render  the  result  less  exact  than  was 
desired.  Chief  among  these  causes  was  the  difficulty  of 
measuring  the  angular  distance  between  the  planet  and  a 
neighboring  star  in  the  nianner  we  have  described.  This 
difficulty  arises  from  the  fact  that  Mars  appears  in  the 
telescope  as  a  disk,  while  the  stars,  of  course,  show  only 
tiny  points  of  light :  and  there  seems  to  be  some  kind  of 
personal  error  introduced  by  the  effort  to  measure  from  a 

266 


THE   SOLAR  PARALLAX 

disk  to  a  point.  For  this  reason,  Gill  decided  to  repeat 
the  work,  using  certain  of  the  planetoids  (p.  231)  whose 
orbits  are  located  between  Mars  and  Jupiter.  These  little 
planetoids,  of  course,  appear  in  the  telescope  like  star 
points,  and  the  above  cause  of  personal  error  does  not 
arise. 

Having  been  appointed  director  of  the  great  observatory 
maintained  by  the  British  admiralty  at  the  Cape  of  Good 
Hope,  Gill  attacked  the  problem  on  a  large  scale,  using  three 
different  planetoids,  Iris,  Sappho,  and  Victoria,  all  of  which 
have  orbits  suitably  situated  for  the  purpose.  He  caused 
to  be  constructed  a  special  instrument  for  measuring  the 
angular  distances  between  the  planetoids  and  neighboring 
small  stars.  This  instrument  is  called  a  Heliometer.  Four 
were  made,  and  mounted  respectively  at  the  Cape  of  Good 
Hope,  New  Haven,  Leipzig,  and  Gottingen.  With  all 
these  special  instruments  simultaneous  observations  were 
made  in  such  a  way  that  both  the  diurnal  and  the  two-ob- 
servatory methods  could  be  used  in  the  subsequent  calcula- 
tions. The  final  result  of  the  whole  campaign  was  to  fix  the 
solar  parallax  at  8. "80 :  this  value  is  now  regarded  as  the 
best,  and  has  been  adopted  by  all  authorities  to  determine 
the  scale  of  the  solar  system,  and  to  perform  calculations  of 
every  kind  relating  to  planetary  motions. 

Since  this  work  of  Gill's  was  completed,  a  certain  newly 
discovered  planetoid,  Eros  (p.  236),  was  found  to  have 
an  orbit  so  placed  that  it  can  at  times  approach  the  earth 
nearer  than  any  other  object  in  the  heavens  except  our  own 
moon.  Consequently,  its  distance  from  the  earth  must 
admit  of  very  accurate  determination.  One  of  the  close 
approaches  of  Eros,  or  favorable  oppositions  (cf.  p.  263), 
occurred  in  1900 ;  and  extensive  observations  were  then  made, 

267 


ASTRONOMY 

this  time  by  the  newly  perfected  method  of  photographic 
observation.  Results  have  been  published  only  very  re- 
cently, and  they  confirm  Gill's  value,  8. "80. 

This  method  of  minor  planet  observation  is  so  superior 
that  all  other  methods  are  of  historic  interest  only;  but, 
historically,  the  famous  determinations  from  transit  of 
Venus  observations  (p.  221)  are  well  worth  a  careful  study. 
If  we  consider  the  motion  of  an  inferior  planet  like  Venus,  and 
assume  the  orbits  of  both  Venus  and  the  earth  to  lie  in  a 
single  plane,  then,  at  each  inferior  conjunction,  Venus  will 
pass  between  the  earth  and  the  sun.  This 
is  evident  from  Fig.  72,  which  once  more 
shows  Venus  at  inferior  conjunction.  In 
point  of  fact,  the  orbits  do  not  lie  exactly  in 
a  plane,  but,  nevertheless,  a  passage  between 
us  and  the  sun  does  sometimes  occur.  It 

FIG.   72.     Venus  at 

inferior  Conjunc-  will  happen  whenever  the  inferior  conjunc- 
tion takes  place  at  about  the  time  when 
Venus  is  at  one  of  the  nodes  (p.  200) ;  or,  in  other  words, 
when  the  conjunction  happens  while  Venus  is  on  the  line  of 
intersection  of  the  two  orbit  planes  of  Venus  and  the  earth. 
When  on  this  line  of  intersection,  Venus  is  for  the  moment 
in  both  planes ;  and  if  there  is  also  an  inferior  conjunction 
at  the  same  time,  there  must  be  a  transit. 

Venus,  during  transit,  is  seen  as  a  small  round  black  dot 
projected  on  the  bright  disk  of  the  sun.  This  dot  appears 
to  enter  the  solar  disk  on  the  western  edge,  transits  the 
sun  in  a  line  approximately  straight,  and  finally  passes  away 
from  the  sun  again  at  the  eastern  edge.  It  then  disappears  ; 
for,  of  course,  we  cannot  see  Venus  against  the  sky  back- 
ground, when  near  the  sun,  since  the  illuminated  side  of  the 
planet  is  then  turned  toward  the  sun.  We  see  the  dark  side  ; 

268 


THE   SOLAR   PARALLAX 

it  is  "new  Venus,"  if  we  may  borrow  a  term  from  the  analogy 
of  the  moon. 

Halley,  a  famous  astronomer  royal  of  England,  showed 
how  to  determine  the  solar  parallax  from  transit  of  Venus 
observations.1  His  method  is  shown  in  Fig.  73. 


FIG.  73.    Transit  of  Venus,  Halley 's  Method. 
(After  HerschePs  Outlines  of  Astronomy,  London,  1851,  p.  289.) 

Two  observers  on  the  earth,  located  at  the  points  A  and  B, 
widely  separated  in  latitude,  are  provided  with  good  clocks, 
and  observe  the  exact  quantity  of  time  required  by  Venus 
to  complete  a  transit  across  the  sun's  disk.  But  these  two 
observers  will  see  Venus  crossing  the  sun  along  two  different 
lines  or  " chords"  SP  and  sp.  The  lengths  of  these  two 
chords  can  be  calculated  in  seconds  of  arc ;  and  thence  the 
solar  parallax  can  be  determined.2 

Halley  thought  it  would  be  possible  to  observe  the  dura- 
tions belonging  to  the  two  chords  within  two  seconds  of 
time.  In  actual  observable  transits  of  Venus  this  would  give 
the  solar  parallax  correct  within  one-hundredth  of  a  second 
of  arc.  But  no  such  precision  of  observation  has  ever  been 
possible,  principally  because  Venus  has  an  atmosphere 
(p.  220)  which  introduces  errors  in  the  observed  durations 
of  the  chords.  Even  in  the  modern  transits  of  1874  and 
1882,  these  errors  were  as  great  as  10  seconds  of  time. 

1  See  Phil.  Trans.  Roy.  Soc.  Lond.,  Vol.  XXIX,  p.  1716,  or  Button's 
Abridgment,  Vol.  VI,  p.  243. 

2  Note  33,  Appendix. 


ASTRONOMY 

There  is  an  interesting  astronomical  story  connected  with 
the  first  observed  transit  of  Venus.  It  seems  that  in  the  year 
1639  there  was  a  young  curate  in  England ;  a  man  living  in 
miserable  circumstances,  but  nevertheless  inspired  with  an 
extraordinary  zeal  in  the  study  of  astronomy.  This  man, 
Horrocks  by  name,  and  at  the  time  but  22  years  of  age, 
united  in  his  own  person  two  of  the  most  poorly  paid  pro- 
fessions, that  of  an  unbeneficent  clergyman  of  the  estab- 
lished church,  and  an  astronomer.  A  diligent  student 
of  Kepler's  writings,  he  had  been  able  to  correct  an  error  of 
the  latter,  and  to  predict  that  a  transit  of  Venus  would 
occur  on  Nov.  4,  1639.  He  was  unable  to  fix  the  exact 
hour,  but  he  had  a  little  telescope,  and  prepared  him- 
self to  watch  the  sun  through  the  entire  day. 

Now  comes  the  peculiarly  human  part  of  the  story.  He 
found  that  the  eventful  date  would  occur  on  a  Sunday.  It 
seemed  of  the  last  importance  to  secure  the  observation, 
which  was  at  that  time  an  unprecedented  one ;  the  circum- 
stances therefore  found  him  undecided  between  his  duty  at 
church  and  his  keen  desire  to  secure  fame  as  an  astronomer. 
His  sense  of  duty  prevailed :  he  decided  to  give  to  the  tele- 
scope only  the  intervals  between  services.  And  he  had  his 
reward,  after  afternoon  service.  Hurrying  to  his  poor  home, 
he  was  in  time  to  see  the  black  round  planetary  dot  on  the 
sun  just  before  sunset,  which  happens  at  a  very  early  hour 
in  the  northern  latitude  of  England,  and  in  November. 
To-day,  a  tablet  may  be  seen  in  Westminster  Abbey, 
bearing  a  Latin  inscription  commemorating  this  famous 
observation.1 

1 A  very  good  account  of  it  is  to  be  found  in  Cassini's  Elements 
d' Astronomie,  published  in  1740.  At  that  time  the  Horrocks  observa- 
tion was  still  unique,  and  Cassini  founds  many  calculations  upon  it.  The 
tablet  in  Westminster  bears  a  quotation  from  Horrocks'  own  work,  Venus 

270 


THE   SOLAR  PARALLAX 

Having  thus  outlined  several  methods  of  determining 
the  sun's  parallax  by  observations  of  planets,  it  remains 
to  mention  certain  indirect  methods  of  arriving  at  its  value. 
For  instance,  the  sun's  distance  can  be  computed  from 
the  theory  of  the  aberration  of  light  (p.  136).1  Another 
quite  independent  way  is  called  the  perturbation  method. 
Briefly  stated,  it  consists  in  measuring  the  slight  perturba- 
tions (p.  403)  produced  in  the  regular  elliptic  motions  of 
the  planets.  These  perturbations  are  caused  by  gravita- 
tional attractions  between  the  bodies  concerned,  and  the 
mathematical  equations  expressing  them  involve  the  distance 
from  earth  to  sun  as  a  factor.  If  this  distance  is  known,  it 
is  possible  to  compute  the  perturbations ;  or,  the  perturba- 
tions being  measured  observationally,  the  solar  distance  may 
be  computed. 

in  Sole  Visa,  published  by  Hevelius  in  1662  at  Dantzig.     The  quotation 
reads,  "Ad  majora  avocatus  quae  ob  haec  parerga  negligi  non  decuit." 
1  Note  34,  Appendix. 


271 


CHAPTER  XV 

ASTRONOMIC    INSTRUMENTS 

IT  is  not  easy  to  understand  the  details  of  instruments 
from  printed  descriptions  and  illustrations.  A  short  verbal 
explanation,  by  an  astronomer  in  an  observatory,  with  the 
instrument  under  discussion  before  him,  is  the  very  best  way 
to  gain  an  insight  into  the  methods  and  machinery  of  obser- 
vation. For  those  who  have  no  opportunity  to  visit  an 
observatory,  we  give  here  a  brief  account  of  the  most  impor- 
tant kinds  of  astronomic  apparatus,  prefacing  it  with  Plate  12, 
a  photograph  of  the  famous  Lick  observatory  buildings  on 
the  summit  of  Mt.  Hamilton,  in  California. 

To  begin  with  the  telescope  itself.  In  the  popular  imag- 
ination, it  is  a  big  tube  more  or  less  filled  with  lenses  from 
end  to  end.  But  this  notion  is  quite  wrong.  Theoretically, 
the  telescope  has  two  lenses  only,  one  at  each  end  of  the  tube. 
The  large  lens,  which  is  turned  toward  the  sky,  is  called 
the  Object  Glass :  upon  it  falls  the  light  coming  from  the 
celestial  object  under  observation.  This  light  is  concen- 
trated or  " focused"  by  the  object  glass,  and  forms  an 
image  of  the  celestial  body  near  the  small  end  of  the  tube 
where  the  observer  places  his  eye.  Between  this  focal 
image  and  the  object  glass,  the  tube  is  empty. 

The  other  telescope  lens  is  placed  at  the  small  end  of  the 
tube,  between  the  observer's  eye  and  the  focal  image,  but 
very  near  the  latter.  This  lens  is  simply  a  magnifying  glass, 
or  microscope,  and  is  intended  to  enlarge  the  focal  image,  so 

272 


ASTRONOMIC   INSTRUMENTS 

that  the  observer  will  see  more  detail  than  would  be  possible 
with  the  eye  alone.  This  eye-end  lens  of  the  telescope  is 
called  the  Eye-piece.  In  modern  instruments,  both  telescope 
lenses  are  of  the  kind  called  " Compound"  lenses.  Each 
is  made  up  of  two  or  three  separate  lenses,  placed  close 
together,  or  even  in  actual  contact.  By  this  compounding 
of  the  lenses  it  has  been  found  possible  to  eliminate  partially 
certain  optical  imperfections  from  which  all  lenses  suffer. 
But  each  compound  lens  really  acts  like  a  simple  lens,  except 
that  it  does  its  work  better.  Galileo's  telescope  of  1610, 
which  found  the  moons  of  Jupiter  and  the  spots  on  the  sun, 
had  single  lenses  only. 

Telescopes  intended  for  terrestrial  use  have  an  extra  lens  in 
the  eye-piece,  called  an  " erecting"  lens.  For  the  simple 
astronomic  telescope  reverses  the  image  of  the  object  we  look 
at ;  we  see  it  with  the  top  and  bottom  interchanged,  and  the 
right  and  left  sides  likewise  inverted.  This  is  of  no  conse- 
quence in  astronomy,  since  there  is  no  up  or  down  in  space, 
and  a  round  planet  may  be  observed  just  as  well  one  way  as 
the  other.  But  for  terrestrial  purposes,  we  must  have  ob- 
jects represented  to  the  eye  in  their  true  positions.  This 
extra  erecting  lens  diminishes  slightly  the  efficiency  of  the 
telescope,  because  it  introduces  two  additional  glass  surfaces, 
the  two  sides  of  the  erecting  lens  itself.  And  as  human 
hands  cannot  grind  lenses  with  absolute  accuracy  to  their 
correct  theoretic  shape,  it  follows  that  the  erecting  lens 
causes  slight  errors  that  do  not  exist  in  the  astronomical 
eye-piece.  Plate  3,  p.  17,  shows  the  moon  as  seen  in  an 
astronomical  or  inverting  telescope  (cf.  p.  166). 

The  question  is  often  asked:  "What  is  the  magnifying 
power  of  a  given  telescope?"  Or  the  same  question  occurs 
in  another  form:  "How  near  does  this  telescope  bring  the 
T  273 


ASTRONOMY 

moon?"  These  two  questions  are  really  one.  The  moon's 
distance  is  240,000  miles  (p.  169) ;  a  telescope  magnifying 
1000  times  would  therefore  bring  it  within  a  distance  of  240 
miles ;  or  give  us  as  good  a  view,  approximately,  as  we  would 
get  with  the  unaided  eye  if  the  moon  were  only  240  miles 
away. 

From  what  has  been  said  before,  it  is  perfectly  clear  that, 
within  certain  limits,  the  magnifying  power  of  a  telescope 
is  just  as  great  as  we  care  to  make  it.  The  magnifying  power 
comes  from  the  power  of  the  eye-piece  regarded  as  a  mi- 
croscope :  it  is  evident  that  we  can  use  a  microscope  of  high 
power,  or  one  of  low  power,  on  the  same  telescope,  at  dif- 
ferent times;  and  thus  we  can  vary  the  magnification  af- 
forded by  the  instrument  as  a  whole.  But  there  is  a  definite 
practical  difficulty  that  limits  the  available  power  of  the 
eye-piece. 

Suppose  we  are  observing  a  planet,  such  as  Mars,  for  in- 
stance. The  quantity  of  light  received  from  Mars  may  be 
regarded  as  constant,  and  therefore  a  constant  quantity  of 
light  from  Mars  reaches  the  focal  image.  This  light  is  there 
spread  uniformly  over  the  surface  of  that  image.  Now  if 
we  double  the  magnifying  power  of  the  eye-piece,  we  shall 
see  twice  as  large  an  image.  The  same  quantity  of  light 
from  Mars  is  therefore  spread  over  a  larger  surface,  and  so 
the  image  is  dimmer  than  before.  Increase  of  magnifying 
power  in  the  eye-piece  enlarges  the  image  and  brings  out 
more  detail;  but  it  makes  that  detail  fainter  (cf.  p.  230). 

If  we  continue  to  increase  the  magnification,  there  must 
come  a  time  when  we  shall  increase  the  detail,  but  will  be 
unable  to  see  it  on  account  of  faintness.  For  these  reasons, 
astronomic  telescopes  are  provided  with  a  " battery"  of 
eye-pieces  of  different  powers.  It  is  customary  for  the 

274 


ASTRONOMIC  INSTRUMENTS 

astronomer  to  try  gradually  increasing  powers  until  he  finds, 
by  experiment,  the  one  that  gives  the  best  result.  It  will  not 
be  the  same  one  each  night,  because  a  little  higher  power 
than  usual  may  be  employed  when  the  terrestrial  atmosphere 
is  especially  clear. 

As  soon  as  the  above  limit  of  power  is  reached,  no  further 
increase  is  possible,  unless  we  enlarge  the  object  glass;  or, 
in  other  words,  make  the  whole  telescope  bigger.  With  a 
larger  object  glass  we  can  gather  more  light  from  Mars, 
because  the  "  light-gathering "  power  of  an  object  glass 
must  increase  with  an  increased  area  of  the  glass  itself. 
And  if  we  gather  more  light,  we  can  have  a  larger  focal 
image  without  making  it  too  dim  for  practical  use.  Ex- 
perience has  shown  that  under  the  most  extremely  favorable 
terrestrial  atmospheric  conditions,  it  is  possible  to  use  a 
magnification  of  about  100  for  each  inch  in  the  diameter  of 
the  object  glass.  In  the  case  of  the  great  Yerkes  telescope, 
the  diameter  of  that  glass  is  40  inches;  a  power  of  4000 
should  therefore  be  conceivably  possible;  and  the  moon 
should  be  brought  within  the  equivalent  of  -!$$$-,  or  60 
miles.  But  this  theoretic  result  is  never  quite  attained  in 
practice,  because  all  imperfections  of  the  atmosphere  are 
magnified  by  increased  optical  power.  We  see  the  moon  as 
we  would  see  it  with  the  unaided  eye  at  a  distance  of  60  miles ; 
but  through  an  atmosphere  more  like  water  than  air. 

Among  the  most  important  accessories  of  a  telescope,  when 
it  is  to  be  used  for  accurate  measurement,  is  a  pair  of  "  cross- 
threads"  at  the  focus.  These  threads  are  usually  made  of 
spider  web ;  for  they  must  be  extremely  delicate,  so  that 
magnification  by  the  eye-piece  will  not  prevent  accuracy  of 
observation.  The  field  of  view  of  a  telescope  provided  with 
cross-threads  would  look  like  Fig.  74.  When  the  focal  image 

275 


ASTRONOMY 

of  a  star  is  brought  to  the  exact  intersection  of  the  cross- 
threads,  by  moving  the  telescope,  the  instrument  is  aimed  or 
" pointed"  accurately  at  the  star;  and  if  the  telescope  is 
attached  to  brass  circles  divided  into  degrees 
and  minutes,  it  is  possible  to  measure  the  exact 
direction  in  which  we  see  the  star  projected  on 

FIG.  74.     Cross-  . 

threads  in  a    the  celestial  sphere.     This  kind  of  measure- 
ment  is   fundamental   in   the    astronomy   of 
precision  (cf.  p.  197). 

Sometimes  the  single  pair  of  fixed  cross-threads  is  replaced 
by  a  pair  of  parallel  threads  aaf  and  W  shown  in  Fig.  75, 
together  with  a  cross-thread  cc' '.  In  such  an  arrangement 
the  two  parallel  threads  aaf  and  W  are  made  movable,  while 
cc'  is  fixed.  The  two  parallel  threads  can  be  moved  nearer 
together  or  farther  apart  by  means  of  suitable  screws  out- 
side the  tube  of  the  telescope ;  and  a  method  is  also  provided 
for  measuring  accurately  their  distance  asunder.  This  ar- 
rangement is  called  a  micrometer  (p.  265),  and  with  it  short 
angular  distances  on  the  sky  can  be  measured  with  very  high 
precision.  Examples  of  such  measurements  are  the  obser- 
vation of  Mars  to  ascertain  the  solar  parallax 
(p.  265),  the  angular  diameter  of  the  sun  (p.  118) 
or  of  the  planets  (p.  203),  etc. 

The  methods  of  mounting  telescopes  for  as- 

.    .     .  ,.  T-    •        FIG.  75.     The 

tronomical  purposes  are  most  interesting.     It  is     Micrometer. 
of  course  essential  that  the  tube  be  movable  :   it 
must  be  possible  to  turn  it  about  pivots  or  "axes,"  in  order 
that  it  may  be  pointed  toward  different  parts  of  the  sky. 
The  most  simple  form  of  mounting  is  indicated  in  Fig.  76, 
which  shows  an  instrument  called  a  Meridian  Circle.     OE 
is  the  telescope,  0  being  the  object-glass  and  E  the  eye-piece. 
At  /  is  the  focus,  containing  the  cross-threads.     AX  is  a 

276 


ASTRONOMIC   INSTRUMENTS 


rotation  axis,  firmly  attached  to  the  telescope.  There  is  no 
motion  of  the  instrument,  except  rotation  around  this  one 
axis ;  but  a  complete  rotation  about  that  axis  is  possible. 
The  line  AX  is  made  to  point  due  east  and  west  when  in 
proper  adjustment;  and 
it  is  made  perfectly 
level.  So  the  telescope 
must  point  north  or 
south  accurately,  since 
it  is  placed  at  right 
angles  to  the  axis  AX. 
It  follows  that  if  the 
"  sight-line  "  EfO  be  con- 
tinued outward  indefi- 
nitely beyond  0,  until 
it  reaches  the  celestial 
sphere,  it  will  meet  that 
sphere  at  some  point  of 
the  celestial  meridian 


(p.  36).  And  if  the 
telescope  is  turned 
through  a  complete 
rotation  around  the 


FIG.  76.     Meridian  Circle. 


axis  AX,  the  sight-line  EfO  may  be  imagined  to  trace  out 
the  celestial  meridian  on  the  sky. 

It  results  from  these  considerations  that  the  meridian 
circle  can  observe  stars  on  the  celestial  meridian  only; 
and,  conversely,  if  a  star  be  observed  on  the  cross-threads  of 
the  telescope,  the  observer  knows  that  it  is  at  that  moment 
projected  on  the  celestial  meridian.  If  the  exact  time  of  the 
observation  be  noted,  too,  it  is  possible  to  calculate  the  right- 
ascension  (p.  34)  of  the  star  ;  and  thus  are  the  right-ascen- 

277 


ASTRONOMY 

sions  of  stars  and  planets  determined  observationally  by 
astronomers. 

This  meridian  instrument  is  provided  also  with  two  brass 
circles  c  and  c',  divided  into  degrees  and  minutes  of  arc. 
By  the  aid  of  these  circles  it  is  possible  to  measure  the  altitude 
(p.  36)  of  the  star,  or  its  angular  elevation  above  the  horizon 
at  the  moment  when  it  is  observed  to  be  on  the  meridian. 
From  this  measurement  of  altitude  it  is  possible  to  calculate 
the  star's  declination  (p.  34) .  Thus  the  meridian  circle  makes 
known  both  the  right-ascension  and  declination  of  the  star 
or  planet,  and  these  give  us  its  exact  location  on  the  sky  at 
the  moment  of  observation. 

It  will  be  noticed  that  a  precise  record  of  the  time  of  such 
observations  is  most  essential.  For  this  purpose  astronomers 
employ  " standard"  pendulum  clocks  of  the  most  extreme 
accuracy,  usually  kept  in  vaults  and  air-tight  cases  where 
the  temperature  and  barometric  pressure  are  not  allowed 
to  vary,  so  as  to  produce  inaccuracy  in  the  running  of  the 
clocks.  For  the  actual  record  of  the  time  of  observation, 
the  clock  is  connected  with  an  electric  "  chronograph."  This 
instrument  maintains  an  automatic  record  of  the  running 
of  the  clock  upon  a  sheet  of  paper  attached  to  a  revolving 
brass  drum;  and  upon  this  same  sheet  the  observer  can 
record  electrically  the  instant  of  time  when  he  makes  his 
observation ;  and  he  can  make  this  record  without  his  error 
ever  exceeding  one-fifth  of  a  second  of  time.  By  taking 
the  mean  of  several  observations,  the  average  error  can  even 
be  reduced  below  this  small  amount. 

The  above  process,  as  outlined,  indicates  the  method  of 
observing  stars  of  unknown  location  on  the  sky  in  order  to 
make  known  their  right-ascensions  and  declinations.  But 
the  same  meridian  telescope,  clock,  and  chronograph  can 

278 


PLATE   13.     The  Lick  Telescope. 


ASTRONOMIC   INSTRUMENTS 

be  used  for  the  observation  of  known  stars ;  and  will  then 
furnish  a  check  upon  the  time  indicated  by  the  standard  clock. 
It  is  by  this  latter  process  that  the  astronomic  standard 
clocks  are  kept  correct  in  order  that  the  observatories  may  be 
able  to  distribute  correct  time  telegraphically  for  the  control 
of  the  " regulator"  clocks  that  are  to  be  found  in  most 
jewelers'  shops,  where  people  "set"  their  watches  (cf.  p.  18). 

Having  thus  described  briefly  the  astronomer's  instrument 
of  precision,  the  meridian  circle,  we  must  next  consider  the 
"equatorial"  mounting,  the  arrangement  with  which  almost 
all  ordinary  telescopic  observations  are  made.  This  is 
the  form  of  mounting  usually  fitted  with  a  micrometer  for 
the  measurement  of  small  angular  diameters,  distances,  etc. 
A  photograph  of  a  large  telescope,  mounted  equatorially,  is 
reproduced  in  Plate  13.  This  instrument  is  set  up  at  the 
Lick  observatory;  the  diameter  of  the  object-glass  is  36 
inches ;  and  the  whole  observatory  floor  is  built  like  an  ele- 
vator, so  that  it  can  be  moved  up  and  down,  to  accommodate 
the  observer  when  the  tube  is  directed  to  various  parts  of  the 
sky.  The  supporting  pillar  of  the  telescope  mounting 
passes  down  through  a  hole  in  the  floor,  so  that  the  in- 
strument itself  is  not  disturbed,  when  the  floor  is  raised  or 
lowered.  James  Lick,  donor  of  the  telescope,  is  buried 
under  it. 

The  first  essential  of  such  a  mounting  is  some  form  of 
"universal  joint,"  so  that  the  tube  may  be  aimed  at  any  part 
of  the  sky.  A  single  axis,  such  as  that  of  the  meridian  circle, 
is  not  sufficient.  Accordingly,  in  the  equatorial,  shown 
again  in  Fig.  77,  we  find  two  axes,  A  and  A',  perpendicular 
to  each  other.  The  telescope  can  be  rotated  around  the  axis 
A' ;  and  this  axis  itself,  with  the  telescope  attached,  can  be 
rotated  around  the  axis  A.  A  combination  of  the  two  rota- 

279 


ASTRONOMY 


tions  furnishes  a  universal  motion,  giving  access  to  any 
part  of  the  sky.  By  means  of  these  rotations,  the  tube  can  be 
moved  from  the  position  of  Fig.  77  to  that  of  Plate  13. 

The  axis  A  is  called  the  polar  axis ;  and  the  instrument  is 
so  constructed  and  adjusted  that  this  axis  points  directly 

toward  the  celestial  pole 
(p.  32).  Since  the  stars 
perform  their  apparent 
diurnal  rotations  (p.  33) 
around  that  pole,  it  fol- 
lows that  they  will  seem 
to  perform  them  around 
the  polar  axis  of  the 
equatorial.  This  sim- 
plifies the  use  of  the  in- 
strument; for  a  star 
once  brought  into  the 
field  of  view  of  the  tele- 
scope, we  can  keep  it  there 
by  moving  the  instru- 
ment around  the  polar 
axis  only.  For  the  tele- 
scope must  be  kept  mov- 
ing to  prevent  the  diur- 
nal rotation  of  the  stars 
from  carrying  the  object  under  observation  out  of  the  field  of 
view.  The  necessary  rotation  around  a  single  axis  can  be 
accomplished  automatically  by  means  of  clock-work,  shown 
inside  the  vertical  supporting  pillar  in  Plate  13.  The  as- 
tronomer is  thus  left  free  to  pursue  his  observations  without 
any  further  attention  to  the  telescope.  If  there  were  no 
inclined  polar  axis,  but,  in  its  place,  a  pair  of  axes,  one  verti- 

280 


FIG.  77.     The  Equatorial. 


PLATE   14.     The  Crossley  Reflector. 


ASTRONOMIC   INSTRUMENTS 

cal  and  the  other  horizontal,  this  simple  clock-work  plan 
would  be  impossible. 

The  equatorial  carries  two  circles,  c  and  c',  divided  into 
degrees,  and  attached  to  the  axes  A  and  A'.  With  the  circle 
c'  it  is  possible  to  measure  the  declination  (p.  34)  of  an  object 
in  the  field  of  view ;  and  with  the  circle  c  its  hour-angle  (cf . 
p.  68)  can  be  measured.  And  if  we  wish  to  find  a  known 
object  which  is  invisible  to  the  unaided  eye,  we  have  merely 
to  turn  the  telescope  around  the  two  axes,  until  the  two 
circles  indicate  the  object's  known  declination  and  hour- 
angle,  when  it  will  at  once  appear  in  the  field  of  view. 

The  foregoing  description  of  the  telescope  applies  to  the 
" refractor"  with  its  object-glass  and  eye-piece.  But  there 
is  another  form  of  instrument,  the  " reflector,"  in  which  the 
object-glass  lens  is  replaced  by  a  curved  mirror.  This 
forms  a  focal  image  similar  to  that  given  by  a  lens  ;  and  the 
image  is  again  examined  with  an  eye-piece  lens  as  before. 
Plate  14  shows  the  equatorially  mounted  Crossley  reflector 
of  the  Lick  observatory.  The  polar  axis  is  shaped  like  a 
double  cone ;  and  the  reflector  is  at  the  bottom  of  the  big 
tube.  The  focal  image  is  formed  near  the  top  of  the  tube, 
and  is  there  examined  with  an  eye-piece  shown  in  the  plate. 
It  was  with  this  instrument  that  Keeler  made  his  famous 
photographs  of  spiral  nebulae,  one  of  which  is  shown  in  Plate 
2  (p.  4).  Others  will  be  described  in  a  later  chapter. 

Telescopes  intended  for  astronomic  photography  are 
always  mounted  equatorially,  and  their  clock-work  mech- 
anism is  made  especially  precise.  For  when  it  is  desired 
to  make  long  photographic  exposures,  the  telescope,  which 
takes  the  place  of  a  camera,  must  of  course  be  kept  in  motion, 
so  as  to  neutralize  the  diurnal  rotation  of  the  stars.  If  this 
is  not  done  quite  exactly,  we  obtain  only  a  worthless  "moved 

/  281 


ASTRONOMY 


negative. "  To  be  certain  of  this  essential  element  in  astro- 
photography,  such  telescopes  are  usually  made  with  two 
tubes,  like  a  pair  of  opera-glasses.  The  one  tube  is  a  photo- 
graphic telescope,  the  other  a  visual  one,  provided  with 
cross-threads.  With  this  arrangement,  the  astronomer  can 
watch  an  object  with  the  visual  telescope,  while  it  is  being 
photographed  in  the  other. 

If  the  clock  does  not  move  quite  perfectly,  the  error  will 
at  once  show  itself ;  for  the  object  will  move  away  from  the 
cross-threads  in  the  visual  telescope.  The  slightest  tendency 
to  such  motion  must  be  prevented ;  and  for  this  purpose  the 
clocks  of  such  instruments  are  provided  with  certain  adjust- 
ing screws,  or  other  devices,  with  which  the  astronomer  can 
correct  any  possible  errors  of  the  clock  while  it  is  actually 
running,  and  while  the  photograph  is  being  exposed. 

The  last  important  astronomic  instrument  to  be  described 

here  is  called  the  Spectro- 
scope, of  which  Fig.  78 
shows  the  essential  parts. 
The  reader  will  recall 
that  if  we  look  at  a 
source  of  light  through 
a  glass  prism,  it  will  be 
spread  out  into  a  band 
of  colors,  violet,  indigo, 
blue,  green,  yellow, 
orange,  red.  The  actual  spectroscope  consists  of  a  prism  P 
(or  a  succession  of  several  prisms)  and  two  brass  tubes.  One, 
the  "collimator"  C,  admits  light  to  the  prism.  It  has  a 
narrow  slit  s,  at  one  end,  so  that  the  light  may  enter  as  a 
thin  line,  parallel  to  the  edge  of  the  prism ;  at  the  other  end 
there  is  a  lens,  0,  to  render  the  rays  of  light  parallel.  The 

282 


FIG.  78.    The  Spectroscope. 


ASTRONOMIC   INSTRUMENTS 

other  brass  tube  is  merely  a  little  view  telescope  O'E,  with 
which  to  examine  the  " spectrum"  and  to  magnify  it.  Fre- 
quently it  is  better  to  substitute  for  the  prism  a  glass  plate 
on  which  a  great  number  of  very  close  parallel  lines  have 
been  ruled  with  a  dividing  engine.  Such  a  glass  spreads 
light  into  a  spectrum  similar  to  the  one  obtained  with  a 
prism.  Plate  15  shows  a  spectroscope  attached  to  the  eye- 
end  of  the  big  Lick  telescope.  Three  prisms  are  used  ; .  and 
the  eye-piece  of  the  view  telescope  is  replaced  by  a  plate- 
holder,  so  that  stellar  spectra  may  be  photographed. 

If  we  send  colored  light  into  the  spectroscope,  the  result 
is  as  follows :  If  the  light  is  red,  it  goes  to  the  red  part  of 
the  spectrum,  where  it  belongs,  and  we  there  see  a  bright  red 
line,  which  is  merely  an  image  of  the  spectroscope  slit.  But 
if  we  send  in  white  light,  which  is  really  .a  mixture  of  all  kinds 
of  colored  light,  the  prism  analyzes  it :  the  yellow  part  goes 
to  the  yellow  part  of  the  spectrum,  etc.  We  then  see  the 
colored  "  continuous  spectrum,"  made  up  of  an  enormous 
number  of  slit-images  side  by  side  (see  Plate  16, 1).  But  if 
the  light  comes  from  an  incandescent  gas,  instead  of  a  solid 
or  liquid,  we  see  no  continuous  spectrum,  but  a  series  of 
bright-colored  lines,  or  images  of  the  slit,  variously  located 
throughout  the  spectrum;  and  the  combination  of  lines  is 
different  for  every  different  gas.  We  can  actually  deter- 
mine the  name  of  the  incandescent  gas  from  the  positions  of 
the  lines.  This  is  well  shown  in  Plate  16,  3,  4,  and  5,  for  the 
incandescent  vapors  of  sodium,  hydrogen,  and  potassium. 

But  the  most  singular  thing  of  all  relates  to  the  "  absorp- 
tion" of  light  by  gases;  and  this  is  one  essential  thing  in 
astronomic  spectroscopy.  If  a  beam  of  white  light  is  passed 
through  a  layer  of  gas  or  vapor  before  entering  the  spectro- 
scope, this  vapor  will  sift  out,  and  absorb,  precisely  those 

283 


ASTRONOMY 

light-rays  or  colors  which  the  gas  or  vapor  would  itself 
emit  if  it  were  incandescent,  and  which  would  then  appear  as 
a  bright-line  spectrum  in  a  spectroscope. 

From  the  foregoing  principles  we  shall  find  it  possible  to 
study  the  chemical  constitution,  or  the  nature  of  the  vapor 
existing  in  the  sun  and  stars.  And  there  is  also  another  prin- 
ciple, due  to  Doppler,  by  means  of  which  we  can  obtain  in- 
formation of  still  another  kind.  It  is  a  fact  that  the  spectral 
lines  may  at  times  be  shifted  out  of  their  proper  positions  as 
ordinarily  seen  in  the  spectrum.  We  are  taught  in  the 
science  of  physics  that  red  light  has  comparatively  long 
light-waves ;  violet  light,  the  shortest  waves.  Now  if  a 
source  of  light,  such  as  a  star,  happens  to  be  increasing  its 
distance  from  us  at  the  time  of  observation,  we  shall  receive 
fewer  light-waves  per  second  than  would  be  the  case  if  the 
distance  were  stationary.  But  if  we  receive  fewer  light- 
waves per  second,  they  will  seem  to  be  longer  waves,  and 
therefore  more  like  the  waves  from  red  light.  The  effect 
is  to  shift  the  observable  lines  toward  the  red  end  of  the  spec- 
trum. This  shift  can  be  measured  with  a  micrometer,  and 
from  such  measurements  it  is  actually  possible  to  deter- 
mine the  velocity  with  which  a  star  is  approaching  us  or 
receding  from  us  in  space  (cf.  p.  245). 

The  planet  Venus  has  often  been  observed  to  test  this 
principle ;  and  the  photographed  spectrum  of  Venus  in 
Plate  16,  6,  shows  plainly  the  shift  of  the  spectral  lines. 
The  middle  spectrum  belongs  to  Venus ;  the  two  outer  ones 
are  artificial  spectra  produced  in  the  observatory  for  compari- 
son. Microscopic  measurement  of  this  "  spectrogram,"  as  it 
is  called,  enables  us  to  compute  that  Venus  was  increasing 
its  distance  from  the  earth  at  the  rate  of  eight  miles  per 
second  when  the  spectrogram  was  made.  From  our  knowl- 

284 


PLATE   16.     Various  Spectra. 


ASTRONOMIC   INSTRUMENTS 

edge  of  the  orbital  motions  of  Venus  and  the  earth,  it  is 
possible  to  check  this  result  by  an  entirely  independent  method 
of  calculation ;  and  thus  no  doubt  remains  as  to  the  correct- 
ness of  the  Doppler  principle. 

In  addition  to  the  spectroscope  just  described,  astronomers 
also  employ  a  ' '  slitless ' '  instrument.  This  is  made  by  mount- 
ing one  or  more  prisms  outside  the  object-glass  of  a  telescope, 
prisms  large  enough  to  cover  the  entire  object-glass.  With 
such  an  instrument  it  is  possible  to  photograph  on  a  single 
plate  the  spectra  of  all  stars  in  the  telescopic  field  of  view, 
while  the  slit  spectroscope  will  give  only  one  star-spectrum 
at  a  time.  It  is  therefore  clear  that  the  slitless  instrument  is 
best  for  statistical  researches  intended,  for  instance,  to  classify 
all  spectra ;  but  the  slit  instrument  is  more  accurate,  and 
is  also  the  only  form  permitting  the  measurement  of  line 
shift  in  accordance  with  Doppler's  principle. 


285 


CHAPTER  XVI 

SUNSHINE 

IN  Chapter  XIV  we  have  considered  at  length  the  ancient 
problem  of  determining  the  distance  of  the  sun  from  our 
earth :  let  us  next  attempt  to  describe  the  sun  itself. 
Here  we  meet  something  distinctly  modern  in  the  venerable 
science;  for  almost  all  knowledge  we  have  of  the  sun  is 
knowledge  obtained  during  the  last  hundred  years.  The 
ancients  knew  little  or  nothing  about  it. 

Our  subject  falls  readily  into  two  parts  :  first,  information 
obtained  by  the  use  of  the  spectroscope  (p*.  282) ;  and 
secondly,  investigations  other  than  spectroscopic.  Let  us 
begin  with  the  chemistry  of  the  sun.  We  have  already  had 
an  explanation  (p.  283)  of  the  manner  in  which  gases  absorb 
certain  light-rays  while  passing  through  them,  each  gas  ab- 
sorbing its  own  particular  combination  of  such  rays.  This 
principle,  applied  to  the  sun,  gives  the  following  result. 
The  body  of  the  sun  sends  out  white  light  which  the  spec- 
troscope would  naturally  split  up  into  a  continuous  spec- 
trum (p.  283).  But  before  this  white  light  can  pass  through 
the  outer  gaseous  layers  of  the  sun,  the  absorption  phenom- 
ena take  place ;  and  so  the  regular  solar  spectrum  appears 
in  the  spectroscope  as  continuous,  but  crossed  by  a  vast 
number  of  black  lines.  These  lines  are  simply  the  dark 
places  where  absorption  has  occurred.  Thus,  for  instance,  if 
there  is  iron  vapor  in  the  outer  solar  atmosphere,  we  shall 
see  dark  lines  in  the  solar  spectrum  at  exactly  the  points 

286 


SUNSHINE 

where  we  should  see  bright  lines  (p.  283)  if  we  vaporized  some 
iron  in  the  laboratory,  and  examined  the  spectrum  of  its  light. 
All  these  dark  lines  are  merely  bits  of  nothingness  occupying 
the  places  where  there  should  be  images  of  the  spectroscope 
slit,  if  absorption  were  absent.  Such  are  the  famous 
Fraunhofer  dark  lines,  named  from  their  discoverer. 

Plate  16,  2,  shows  a  number  of  the  principal  ones,  together 
with  the  letters  by  which  spectroscopists  designate  them. 
The  double  ZMine,  for  instance,  arises  from  absorption  due 
to  sodium  vapor,  as  may  be  seen  from  a  comparison  with 
the  sodium  vapor  spectrum,  3. 

So  much  being  premised,  we  can  now  explain  the  method 
of  using  the  spectroscope  to  ascertain  the  sun's  chemical  com- 
position. It  is  merely  necessary  to  attach  a  spectroscope  to 
an  ordinary  telescope  in  such  a  way  that  the  slit  will  be  in 
the  telescopic  focal  plane  (p.  272).  Arrangements  must  then 
be  made,  by  means  of  a  tiny  reflecting  prism  attached  to 
the  slit,  to  throw  into  the  view  telescope  the  spectrum  of 
any  desired  substance  vaporized  and  heated  to  incandes- 
cence in  the  observatory,  near  the  telescope.  We  then  see 
the  solar  spectrum  and  the  artificial  spectrum  side  by  side. 

Now  this  artificial  spectrum  is  a  bright-line  spectrum 
(p.  283) :  and  if  opposite  each  of  its  bright  lines  we  find  a  dark 
Fraunhofer  line  in  the  solar  spectrum,  we  have  conclusive 
proof  that  the  substance  vaporized  in  the  observatory  is 
actually  present  as  a  gas  in  the  outer  atmosphere  of  the 
sun.  Many  terrestrial  chemical  elements  have  been  thus 
found  in  the  sun :  the  general  conclusion  is  that  earth  and 
sun  have  a  similar  constitution,  which  was  to  be  expected, 
if  we  accept  any  hypothesis  postulating  a  common  origin 
for  the  sun  and  earth.  (Cf.  nebular  hypothesis,  p.  235.) 

Next  we  must  consider  a  very  interesting  phenomenon 

287 


ASTRONOMY 

called  the  Reversing  Layer.  We  have  seen  that  the  Fraun- 
hofer  dark  lines  are  due  to  absorption  in  the  outer  gases  of  the 
sun.  But  these  gases  are  themselves  so  hot  as  to  be  incan- 
descent. It  is  only  because  the  inner  sun  is  so  much  hotter 
and  so  much  more  luminous  that  we  ordinarily  see  only  the 
light  from  the  inner  sun  and  not  from  the  outer  gases.  The 
latter  are  dark  by  comparison  only. 

There  is  just  one  occasion  when  it  is  possible  to  observe 
the  light  from  the  outer  gases  separately  and  directly. 
This  occurs  during  a  total  solar  eclipse,  when  the  moon  hap- 
pens to  pass  accurately  in  line  between  the  earth  and  the 
sun.  On  such  an  occasion,  when  the  lunar  globe,  advancing 
in  its  orbit  around  the  earth,  has  almost  covered  the  sun, 
just  before  it  is  covered  absolutely,  there  must  be  a  moment 
when  a  tiny  sickle  of  the  outermost  layer  of  the  sun  is  alone 
visible.  At  that  exciting  moment,  and  at  that  moment  only, 
can  we  look  upon  the  outermost  incandescent  gases. 

But  their  light  suffers  no  further  absorption ;  and  so,  like 
all  incandescent  gases,  should  give  a  spectrum  consisting  of 
bright  lines  only.  And  this  is  precisely  what  occurs.  If  we 
observe  the  advancing  eclipse,  just  for  an  instant  before 
totality,  the  continuous  solar  spectrum  with  its  myriad 
black  Fraunhofer  lines  is  suddenly  replaced  by  a  bright- 
line  spectrum.  Each  bright  line  corresponds  accurately  to 
one  of  the  vanished  dark  lines,  since  the  dark  lines  were 
caused  by  absorption  due  to  the  very  gases  that  are  now 
furnishing  the  bright-line  spectrum.  The  critical  instant 
over,  the  sun  is  covered  totally,  and  the  bright  lines  in  turn 
disappear,  too.  This  phenomenon  is  appropriately  termed 
the  Flash  Spectrum. 

The  surface  we  see  when  we  turn  a  telescope  upon  the  sun 
is  called  the  Photosphere.  It  is  not  uniformly  brilliant, 

288 


PLATE  17.    The  Sun. 


Photo  by  Fox. 


SUNSHINE 

but  shows  certain  brighter  " nodules,"  and  also  especially 
bright  points  called  "faculse,"1  which  appear  mostly  at  the 
edges  of  the  sun  and  near  the  sunspots.  They  are  perhaps 
suspended  in  the  solar  atmosphere  at  a  high  level,  and  owe 
their  extra  brightness  to  our  observing  them  through  a 
somewhat  thinner  layer  of  solar  atmosphere.  Probably 
everything  we  see  when  we  examine  the  sun  is  "  atmosphere." 
Young  aptly  compares  the  state  of  affairs  to  a  gas-burner,  in 
which  the  heated  particles  of  the  mantle  are  far  more  lumi- 
nous than  the  flame  of  gas  which  heats  them. 

Let  us  next  enumerate  some  of  the  principal  facts  known 
about  the  sunspots  (p.  17).  We  are  on  sure  ground  when 
we  speak  of  their  size ;  for  we  can,  as  usual,  measure  their 
angular  diameters,  and  then  compute  their  linear  dimensions 
from  our  knowledge  of  the  sun's  distance.  At  times  they 
are  50,000  miles  in  diameter;  and  exceptionally  large  ones 
can  even  be  seen  without  a  telescope.  But  our  knowledge  is 
less  certain  when  we  attempt  to  explain  their  cause.  They 
are  to  be  regarded  probably  as  solar  atmospheric  disturbances 
or  storms.  In  that  case  we  should  expect  them  to  shift  their 
positions  on  the  sun's  surface,  much  as  storm-centers  move 
across  our  earth.  And  we  find  by  observation  that  all  spots 
have  a  common  drift;  and  those  near  the  solar  equator  also 
drift  toward  it,  while  those  far  from  the  equator  drift  toward 
the  solar  poles.  This  might  be  analogous  to  our  phenomena 
of  the  trade-winds,  especially  as  spots  never  occur  near  the 
solar  poles,  or  exactly  at  the  equator. 

So  the  real  cause  of  the  spots  must  be  regarded  as  un- 
known. They  may  be  eruptions  from  the  interior ;  they 
may  be  gases  rushing  downward  into  hollows.  But  we 
cannot  help  thinking  they  are  vast  storms  of  some  kind; 

1  Plate  17. 
U  289 


ASTRONOMY 

storms  of  which  the  materials  are  incandescent  gases,  moving 
with  great  velocities,  and  at  enormously  high  temperatures. 

The  duration  of  individual  spots  is  not  great,  never  more 
than  18  months ;  and  the  central,  apparently  blackest  part 
of  the  spots,  called  the  " umbra,"  is  not  really  dark,  but 
appears  so  only  though  contrast  with  the  much  more  lumi- 
nous surrounding  solar  material.  They  have  also  a  periodic- 
ity, discovered  in  1843  by  Schwabe ;  and  this  is  perhaps  the 
most  interesting  of  the  many  unexplained  observations  of  the 
sun.  Schwabe  found,  by  constant  watching  of  the  solar 
surface,  that  every  eleven  years  there  is  a  period  of  extra 
great  spot  frequency.  This  discovery  owes  its  importance 
to  the  known  fact  that  there  exists  also  an  eleven-year 
period  in  the  frequency  of  terrestrial  magnetic  storms : 
and  especially  great  sunspots  are  always  accompanied  with 
very  strong  magnetic  disturbances  and  auroral  displays  on 
earth.  This  establishes  the  existence  of  some  intimate 
magnetic  relation  between  earth  and  sun;  but  it  has  not 
yet  been  possible  to  reach  a  satisfactory  explanation  of  it. 
Nor  have  astronomers  been  able  to  make  certain  that  any 
other  terrestrial  meteorological  phenomena  exhibit  a  real 
connection  with  the  spots,  though  many  efforts  have  been 
made  to  do  so  on  account  of  the  assistance  such  investigations 
might  give  in  the  matter  of  weather  prediction. 

The  accompanying  Plate  18  is  a  photograph  of  a  partic- 
ularly large  sunspot  which  occurred  July  17,  1905.  It  had 
an  unusual  and  very  brilliant  " bridge"  across  the  umbra. 

The  next  important  question  requiring  consideration  re- 
lates to  the  size  of  the  sun.  We  have  already  determined  its 
distance  to  be  about  93  million  miles.  Observations  very 
similar  in  principle  to  methods  already  explained  for  the 
moon  and  planets  enable  us  to  measure  the  sun's  apparent 

290 


SUNSHINE 

angular  diameter  (cf .  p.  203) ;  and  this  we  find  to  be  32'  4", 
on  the  average.  In  Fig.  79  we  then  have,  as  usual,  a  long, 
narrow  triangle,  of  which  the  base  is  the  sun's  linear  diameter 
AB.  This  we  can  calculate  readily,  because  we  know  the 
sun's  angular  diameter,  or  the  angle  32'  4"  at  the  vertex 


FIG.  79.    Sun's  Diameter. 

of  the  triangle,  situated  on  the  earth.  The  result  comes 
out  nearly  nine  hundred  thousand  miles  for  the  sun's  linear 
diameter. 

Comparing  this  with  the  known  terrestrial  diameter 
(p.  97),  we  find  that  the  sun's  diameter  is  approximately  110 
times  that  of  the  earth. 

To  ascertain  the  sun's  mass  is  a  little  more  difficult  than  to 
find  its  diameter  ;  but  it  can  be  estimated  by  simple  mathe- 
matical methods,1  which  show  that  it  is  about  330,000 
times  the  earth's  mass. 

We  can  also  calculate  the  force  of  gravity  that  must  exist 
on  the  solar  surface  as  compared  with  the  gravitational  at- 
traction existing  on  our  earth.  For  the  gravity  force  on  the 
surface  of  a  sphere  is,  by  Newton's  law,  proportional  to  the 
mass  of  the  sphere,  divided  by  the  square  of  its  radius.  If 
we  then  consider  all  solar  quantities  expressed  in  terms  of  the 
corresponding  terrestrial  quantities  as  units,  we  have  : 

B  .   --          ,  solar  mass         330000 

Solarforceof  gravity  =  (golar  radiug)  2  =  ~  -  28,  approximately. 


This  means  that  an  ordinary  one-pound  weight  would 


1  Note  35,  Appendix. 
291 


ASTRONOMY 

weigh  28  pounds,  if  transported  to  the  sun's  surface,  and 
there  weighed  with  an  ordinary  terrestrial  spring-balance. 

The  solar  volume  or  bulk  compares  with  that  of  the  earth 
in  the  proportion  of  the  cubes  of  their  radii ;  that  is,  as  1  to 
(HO)3.  This  makes  the  solar  volume  1,300,000  times  the 
earth's.  And  since  density  or  specific  gravity  is  proportional 
to  mass  divided  by  volume,  it  follows  that  the  solar  density, 
as  compared  with  the  earth's,  is  : 

solar  mass          330000 
solar  volume  ~  1300000  "  °'25> 

or  only  about  }  the  earth's  density.  The  latter,  com- 
pared with  water,  is  about  5.5 ;  so  the  solar  density  is  only 
about  1^  times  that  of  water.  This  means  that  a  cubic  foot 
of  average  solar  material,  transported  to  the  earth's  surface, 
would  there  weigh  only  about  1J  times  as  much  as  a  cubic 
foot  of  water. 

The  quantity  of  light  and  heat  received  by  us  from  the  sun 
is  certainly  enormous ;  and  yet  it  cannot  be  more  than  a 
small  fraction  of  the  total  quantity  actually  radiated  into 
space.  A  most  interesting  question  arises  in  connection 
with  this  matter :  How  does  the  sun  maintain  through  the 
ages  so  gigantic  an  output  of  heat  energy?  What  is  the 
source  of  the  sun's  heat  ?  Helmholtz  has  proposed  a 
plausible  possible  cause, — the  shrinkage  of  the  sun's  vast 
bulk  under  the  influence  of  its  own  gravitational  attraction. 
He  computed  that  an  annual  shrinkage  of  only  300  feet 
in  the  solar  diameter  would  be  transformable  into  enough 
heat  energy  to  keep  radiation  active  as  it  now  is.  And  it 
would  require  8000  years  for  this  diminution  of  size  to 
reduce  the  sun's  observable  angular  diameter  by  a  single 
second  of  arc ;  nor  could  any  smaller  diminution  be  discov- 
ered by  observation  with  actual  astronomic  instruments. 

292 


SUNSHINE 

The  fact  that  we  have  not  observed  a  reduction  of  the  sun's 
size  is  therefore  no  argument  against  the  Helmholtz  theory. 

Up  to  this  point  we  have  supposed  the  sun  to  consist  of 
a  highly  heated  interior  of  more  or  less  unknown  constitu- 
tion, surrounded  by  an  atmosphere  of  incandescent  gases 
which  produce  the  Fraunhofer  lines  by  absorption,  and  the 
bright-line  flash  spectrum  during  an  eclipse.  But  we 
know  much  more  than  this.  Beyond  the  photosphere 
and  the  reversing  layer  of  gases  is  the  Chromosphere,  or 
color  sphere,  composed  principally  of  great  flaming  masses  of 
red  hydrogen  vapor.  Sometimes  great  red  jets  burst  out- 
ward to  immense  elevations  from  the  solar  surface.  These 
are  the  Prominences  (Plate  19) ;  and  while  these  various 
solar  layers  have  different  names,  it  must  not  be  supposed 
that  they  are  distinct.  They  intermingle,  doubtless,  at 
their  boundaries,  and  melt  into  each  other  without  sudden 
interruptions. 

The  hydrogen  prominences  were  first  seen  during  a  total 
solar  eclipse,  when  the  photosphere  was  completely  covered 
by  the  moon.  But  just  after  the  eclipse  of  1868,  Janssen  and 
Lockyer  for  the  first  time  succeeded  in  observing  them  with- 
out an  eclipse.  We  cannot  see  them  by  merely  looking  at 
the  sun  with  a  telescope,  and  covering  the  central  part  of 
the  solar  disk  at  the  telescopic  focus,  because  the  terrestrial 
atmosphere  is  strongly  illuminated  by  the  sun  itself,  arid  so 
the  prominences  become  invisible  by  contrast.  But  if  we 
bring  the  slit  of  a  spectroscope  tangent  to  the  sun's  disk  at  the 
focus  of  a  telescope,  and  open  the  slit  wide,  the  prominences 
become  visible. 

For  we  then  see  two  spectra  superposed,  one  upon  the 
other.  The  first  is  an  ordinary  continuous  solar  spectrum 
derived  from  the  diffused  sunlight  in  the  terrestrial  atmos- 

293 


ASTRONOMY 

phere,  the  second  a  bright-line  spectrum  from  the  incandes- 
cent hydrogen  of  the  prominences.  Now,  if  we  employ  in  the 
spectroscope  a  number  of  prisms,  instead  of  a  single  one, 
both  these  spectra  will  be  spread  out  to  a  great  length. 
The  continuous  spectrum  will  be  thereby  rendered  dimmer, 
but  the  bright-line  spectrum  will  have  its  lines  separated 
further,  without  rendering  them  less  brilliant.  If  we  con- 
tinue thus  increasing  the  " dispersion"  of  the  spectroscopic 
prisms,  we  shall  finally  diminish  the  luminosity  of  the  at- 
mospheric continuous  spectrum  until  it  disappears  practi- 
cally, and  we  see  only  the  bright-line  spectrum  of  the  promi- 
nence. 

Now,  as  we  know,  these  bright  lines  are  ordinarily  merely 
images  of  the  slit.  But  if  the  slit  has  been  opened  wide 
enough  to  be  wider  than  the  angular  diameter  of  the  promi- 
nences, the  bright  lines  become  images  of  the  prominences, 
instead  of  images  of  the  slit.  We  have  therefore  merely  to 
point  the  view  telescope  at  a  place  in  the  spectrum  where 
there  is  ordinarily  a  bright  hydrogen  line,  and  we  shall  see 
the  prominence,  if  there  happens  to  be  one  on  the  sun's 
edge  at  the  point  where  we  have  placed  the  widely  open 
slit  tangent  to  the  sun's  image  at  the  telescopic  focus. 

In  1891  Hale  invented  an  instrument  called  a  spectrohelio- 
graph,  with  which  the  prominences  may  be  photographed 
without  an  eclipse.  Plate  19  was  made  with  such  an  instru- 
ment. It  utilizes  the  light  of  calcium  gas,  which,  like  hydro- 
gen, is  plentiful  in  the  prominences ;  and  can  be  made  to 
give  a  fine  bright  line  in  the  middle  of  the  usual  dark  Fraun- 
hofer  calcium  line  due  to  the  photosphere  and  reversing 
layers.  The  spectrum  is  allowed  to  fall  on  a  screen  having  a 
second  narrow  slit  corresponding  accurately  to  the  bright 
calcium  line  from  the  prominence.  Through  this  slit 

294 


SUNSHINE 

the  calcium  light  which  originated  in  the  prominence  passes 
to  a  photographic  plate,  so  that  the  plate  receives  prominence 
light  only.  Now,  by  mechanical  means,  the  original  slit 
of  the  spectroscope  is  moved  across  the  solar  image  at  the 
telescopic  focus ;  and  the  second  slit  in  the  screen  is  moved 
in  unison.  The  result  is  to  build  up  a  picture  of  the  sun  on 
the  photographic  plate  with  light  from  the  outer  solar  layer 
only,  and  thus  to  secure  a  photograph  of  the  prominences. 

Still  another  extraordinary  solar  phenomenon  has  been 
discovered  during  total  eclipses.  This  is  the  Corona,  which 
bursts  into  view  when  the  sun  is  completely  concealed  by 
the  moon,  and  appears  as  a  faint  luminous  ring,  of  more  or 
less  irregular  shape,  around  the  sun.  We  know  that  it 
belongs  to  the  sun,  because  its  spectrum  is  that  of  an  in- 
candescent gas,  not  a  continuous  solar  spectrum,  such  as 
it  would  be  if  we  had  here  to  do  merely  with  solar  light 
reflected  in  some  way  from  the  moon ;  and  also  because  the 
coronal  form  always  appears  the  same  at  the  same  moment, 
even  when  seen  from  observatories  widely  separated  on  the 
earth.  Beyond  this  little  is  really  known  for  certain  as  to 
the  corona. 

Plate  20  is  a  photograph  of  the  corona  during  a  total 
eclipse,  the  sun's  disk  being  entirely  covered  by  the  inter- 
posed moon.  The  form  of  the  streamers  indicates  that  the 
phenomenon  may  be  electric  or  magnetic.  A  large  promi- 
nence appears  near  the  lower  part  of  the  plate,  jutting  out 
from  the  obscured  solar  disk.  The  height  of  this  prominence 
is  estimated  by  comparing  it  with  the  photographed  diameter 
of  the  sun :  this  would  make  the  prominence  about  40,000 
miles  high. 

The  question  of  solar  axial  rotation  can  be  examined  best 
by  studying  the  apparent  motions  of  the  spots.  (Cf.  the 

295 


ASTRONOMY 

case  of  the  planets,  p.  202.)  It  is  found  that  they  always 
seem  to  cross  the  solar  disk  from  east  to  west ;  and  when 
one  of  them  makes  a  complete  rotation,  disappearing  at 
the  western,  and  later  re-appearing  at  the  eastern  edge  of 
the  sun,  the  whole  revolution  takes  about  27J  days.  But 
this  is  not  the  true  period  of  solar  axial  rotation  :  the  above 
observed  period,  and  the  true  one,  are  related  by  an  equation 
analogous  to  the  equation  connecting  the  synodic  and  sidereal 
periods  of  the  planets  (p.  209),  since  we  must  correct  the 
observed  period  for  the  effect  of  the  earth's  orbital  motion 
around  the  sun  during  the  time  occupied  by  the  latter  in 
turning  on  its  axis.1  This  correction  makes  the  true  axial 
rotation  period  of  the  sun  about  25|  days. 

These  sunspot  motions  not  only  tell  us  the  period  of  the 
sun's  axial  rotation;  they  also  enable  us  to  ascertain  the 
direction  in  space  of  the  rotation  axis  (cf.  p.  203).  The 
apparent  paths  of  the  spots  are  generally  curved ;  but  on 
June  3  and  December  5  they  appear  quite 
straight.  Referring  to  Fig.  80,  we  see  that  this 
straightness  determines  on  these  dates  the  axial 
FIG.  so.  Sun's  or  polar  points  A  and  B  on  the  sun's  edge,  and 
also  the  angle  by  which  the  solar  rotation  axis 
is  inclined  to  the  plane  of  the  ecliptic.  The  angle  is  about 
83°. 

1  The  planetary  equation  here  takes  the  form : 

11  1 


true  period  of  rotation       length  of  year       observed  period  of  rotation 


296 


CHAPTER  XVII 

ECLIPSES 

ECLIPSES  are  occasional  phenomena;  they  are  usually 
defined  as  temporary  obscurations  of  the  sun  or  moon, 
either  wholly  or  in  part.  We  have  seen  that  these  two 
bodies  are  visible  from  two  very  different  causes :  the  sun  is 
self-luminous,  —  it  is 
visib  le  because  it  sends  '  ^ 


.  .  1  •      1    i  lourii  \_J 

us  its  own  light  ;   the    y        J  Moon 

moon   is   merely  ren- 
dered visible  when  il- 


luminated by  the  sun.      /^"^N          Lunar    Eclipse 
Therefore     a    solar 


(   &un    ) 

eclipse  can  occur  only  FlG  81    Eclipses. 

if    the    moon     inter- 

poses between  us  and  the  sun,  thereby  preventing  our  see- 
ing it  ;  but  a  lunar  eclipse  happens  when  the  earth  passes 
between  the  moon  and  sun,  so  that  solar  light  cannot  reach 
the  moon,  and  render  it  visible.  Thus  there  is  not  neces- 
sarily any  actual  obstruction  in  the  way  of  our  seeing  the 
eclipsed  moon  ;  it  is  invisible  merely  because  it  is  dark  for 
the  time  being.  Figure  81  makes  all  this  plain. 

Hipparchus  was  the  first  to  explain  eclipses,  and  the 
method  of  making  a  fairly  good  approximate  prediction  of 
their  occurrence.  Figure  81  shows  that  if  the  orbits  of  the 
earth  and  moon  were  both  situated  in  a  single  plane  surface 
(here  represented  by  the  plane  of  the  paper)  there  must 
result  one  eclipse  of  the  sun  and  one  of  the  moon  during 

297 


ASTRONOMY 

each  revolution  of  the  moon  in  its  orbit  around  the 
earth.  This  simple  state  of  affairs  is  modified  and  com- 
plicated by  the  fact  that  the  lunar  orbital  plane  is  actually 
inclined  about  5°  to  the  ecliptic  plane,  in  which  the  earth's 

orbit  is  situated  (p. 
160).  Figure  82  is 
supposed  to  represent 

S  Ecliptic  ,.  * 

FIG.  82.     Moon's  Circle.  a   Portion  of  the  SUr- 

face  of  the  celestial 

sphere.  NS  is  part  of  the  ecliptic  circle,  in  which  the  sun  is 
always  seen ;  and  NM  is  part  of  the  great  circle  cut  out  on 
the  sphere  by  the  plane  of  the  moon's  orbit,  in  which  great 
circle  the  moon  is  seen,  for  the  same  reason  that  the  sun  is 
always  seen  in  the  ecliptic  (p.  160).  N  is  the  point  of  inter- 
section of  these  two  great  circles  on  the  celestial  sphere,  the 
angle  between  them  being  5°  only.  The  point  N  is  called 
the  "node"  of  the  lunar  orbit;  and  there  is  another  similar 
node  on  the  opposite  side  of  the  sky,  because  any  pair  of 
great  circles  must  necessarily  intersect  at  two  opposite 
points  on  the  sphere. 

We  have  already  seen  that  the  moon  moves  around  the 
earth,  and  therefore  appears  to  travel  around  the  sky  among 
the  stars,  at  the  rate  of  about  13°  per  day,  so  that  it  overtakes 
and  passes  the  sun  once  in  each  synodic  period  or  lunar 
"month"  (p.  161).  When  it  thus  passes  the  sun,  the  two 
bodies  are  said  to  be  in  conjunction  (cf.  p.  209).  If  this 
conjunction  happens  to  occur  exactly  at  the  nodal  point  N, 
then  sun,  moon,  and  earth  will  lie  in  a  single  straight  line, 
and  a  "central"  eclipse  will  occur.  Furthermore,  it  will 
be  an  eclipse  of  the  sun;  for  if  the  sun  and  moon  appear 
projected  at  a  single  point  of  the  sky,  they  must  both  lie 
on  the  same  side  of  the  earth  (Fig.  81,  solar  eclipse). 

298 


ECLIPSES 

But  an  eclipse  can  also  happen  when  the  sun  and  moon 
are  in  opposition,  or  180°  apart,  as  seen  projected  on  the 
sky.  If  such  an  opposition  takes  place  when  the  moon  is 
exactly  in  the  node  N,  and  the  sun  in  the  other,  or  opposite, 
node,  we  once  more  have  sun,  earth,  and  moon  in  a  single 
straight  line,  and  a  central  eclipse  takes  place.  Only,  in 
this  case,  the  earth  is  between  the  sun  and  moon  (Fig.  81, 
lunar  eclipse),  and  the  central  eclipse  is  a  lunar  eclipse. 

The  problems  connected  with  eclipse  prediction  would  pre- 
sent but  little  of  interest  beyond  the  above,  if  the  node  N 
always  remained  at  the  same  point  of  the  ecliptic.  But  this 
node  is  constantly  moving  along  the  ecliptic, — a  phenomenon 
somewhat  analogous  to  precession  of  the  equinoxes  (p.  126) 
in  the  case  of  the  earth.  Only,  the  lunar  node,  unlike  the 
equinoxes  of  the  terrestrial  orbit,  moves  quite  rapidly, 
making  a  circuit  of  the  entire  ecliptic  once  in  about  19  years. 
It  is  this  phenomenon  that  complicates  the  eclipse  problem 
and  makes  it  interesting. 

Up  to  this  point  we  have  supposed  eclipses  to  occur  at 
the  exact  nodes  only ;  and  this  would  be  the  case  if  the  sun 
and  moon  appeared  to  us  on  the  sky  as  mere 
mathematical  points,  like  the  fixed  stars. 
But  both  sun  and  moon  have  quite  large 
disks,  as  we  see  them  projected  on  the  sky. 
Each  disk  has  a  diameter  of  about  half  a 
degree  of  arc.  Consequently  (Fig.  83),  when  FIG.  83.  Contact 
a  conjunction  occurs,  these  two  disks  may 
just  touch  if  their  centers  are  half  a  degree  apart,  provided 
we  suppose  the  terrestrial  observer  located  at  the  earth's 
center. 

But  an  observer  on  the  surface  of  the  earth,  as  at  the  point 
0  in  Fig.  84,  will  see  the  lunar  and  solar  disks  in  contact  when 

299 


ASTRONOMY 

the  moon  is  at  MI  ;  while  to  an  observer  at  the  earth's 
center  c,  there  would  be  no  contact  until  the  moon  had 
advanced  to  M2.  It  is  clear  from  Fig.  83  that  the  angle 
ScM2  is  |°.  The  angle  ScMi  (from  the  center  of  the  sun 

to  the  center  of  the  moon,  as 
seen  from  the  center  of  the 
earth)  is  1  J°,  approximately,1 


Earth     thus   enlarging    greatly   the 

FIG.  84.     Observer^the  Surface  of  the    possibility  of  an  eclipse  being 

actually  visible  from  some 
point  or  other  on  the  earth's  surface. 

Now  referring  again  to  Fig.  82,  we  see  that  if  the  conjunc- 
tion occurs  when  the  centers  of  the  sun  and  moon  are  at 
the  points  S  and  M,  just  far  enough  from  the  node  A^  to 
make  these  two  points  1J°  apart,  the  two  disks  will  just 
touch,  and  we  shall  barely  escape  the  occurrence  of  an  eclipse, 
visible  from  some  point  of  the  earth's  surface.  If  M  and  S 
happen  to  be  a  little  nearer  the  node,  the  two  disks  will 
overlap,  and  there  must  be  at  least  a  partial  solar  eclipse. 

Knowing  the  angle  at  N  to  be  5°,  there  is  no  difficulty  in 
calculating  how  great  must  be  the  angular  distances  NM  and 
NS,  to  make  the  distance  SM  just  1|°.  This  distance  NM 
is  thus  found  to  be  about  17° ;  so  that  when  conjunction 
occurs  within  about  17°  of  the  node,  there  is  an  eclipse. 
But  this  number  17°  may  vary  all  the  way  from  15°  to  19° 
in  different  years,  largely  on  account  of  small  periodic 
changes  of  the  angle  N,  between  the  two  orbital  planes. 
These  changes  are  of  course  due  to  orbital  perturbations 
(p.  206).  The  number  17°  is  called  the  "solar  eclipse  limit." 

As  we  have  seen,  these  eclipses  at  conjunction  are  solar 
eclipses;  but  corresponding  eclipse  limits  exist  also  in  the 

1  Note  36,  Appendix. 
300 


ECLIPSES 

case  of  oppositions  of  the  sun  and  moon,  when  lunar  eclipses 
occur.  Referring  again  to  Fig.  81,  we  see  that  a  solar  eclipse 
is  possible,  only  if  the  disks  of  the  sun  and  moon,  as  projected 
on  the  sky,  actually  overlap.  But  Fig.  85  makes  plain  that 
a  lunar  eclipse  will  happen  if  the  moon  enters  or  touches  the 
shadow  cast  into  space  by  the  earth.  But  the  apparent 
angular  diameter  of  this  shadow,  as  seen  from  the  earth, 


FIG.  85.     Lunar  Eclipse. 

and  at  the  distance  of  the  moon  from  the  earth,  is  larger 
than  the  moon's  angular  diameter.  The  result  is  that  a 
lunar  eclipse  takes  place  if  the  center  of  the  moon  and  the 
center  of  the  shadow  are  separated  by  an  angular  distance 
of  less  than  about  1°,  as  seen  from  the  earth.  We  can,  of 
course,  again  calculate  how  far  the  sun  must  be  from  the 
node,  at  the  time  of  opposition,  to  make  this  angular  distance 
less  than  1°.  We  thus  find  the  lunar  eclipse  limit  of  about 
11°,  with  a  variation  between  10°  and  12°,  in  round  numbers. 
Since  conjunction  always  happens  at  new-moon,  and  op- 
position at  full-moon,  it  follows  from  the  foregoing  simple 
considerations  that  there  will  be  a  solar  eclipse  at  the  time 
of  new-moon,  if  the  sun  is  within  about  17°  of  the  node; 
and  there  will  be  a  lunar  eclipse  at  the  time  of  full-moon  if 
it  is  within  about  11°  of  the  node.  Two  other  simple  conclu- 
sions follow  at  once  :  (1)  Since  the  sun  appears  to  move  about 
1°  daily  in  the  ecliptic,  there  will  be  a  solar  eclipse  if  the  date 
of  new-moon  falls  within  17  days  of  the  date  when  the  sun 
appears  in  the  node.  And  there  will  be  a  lunar  eclipse  if 
the  date  of  full-moon  falls  within  11  days  of  the  same  date, 
all  in  round  numbers.  (2)  The  solar  eclipse  limit  being 

301 


ASTRONOMY 

the  larger,  solar  eclipses  must  be  more  frequent,  on  the  whole, 
than  lunar  eclipses. 

Hipparchus  was  the  first  to  explain  correctly  the  seeming 
paradox  that  lunar  eclipses  are  seen  much  more  often  than 
solar  eclipses,  although  the  latter  occur  more  frequently. 
The  reason  is  perfectly  simple.  When  the  earth  interposes 
between  the  sun  and  moon,  and  the  moon  thus  enters  the 
earth's  shadow,  it  becomes  dark  at  once,  because  it  gives  no 
light  of  its  own.  Consequently,  any  one  on  the  earth  who 
should  be  able  to  see  the  moon  will  fail  to  see  it  on  account 
of  the  eclipse.  But  at  any  given  instant,  the  moon  should 
be  visible  from  half  the  earth's  surface ;  therefore,  if  there 
is  a  lunar  eclipse,  at  least  half  the  earth's  inhabitants  will 
see  it. 

But  in  the  case  of  a  solar  eclipse,  the  sun  is  not  made  dark. 
The  sun's  light  is  actually  cut  off  by  the  interposed  moon ; 
and  it  is  never  at  any  one  time  thus  cut  off  from  observers 

living  in  more  than  a  small 
part  of  the  earth's  surface.  In 
Fig.  86,  an  observer  on  the 
earth  at  A  will  see  the  moon 
projected  in  his  zenith,  just  as  it 
would  be  seen  by  an  observer 
at  the  earth's  center  C.  But 

FIG.  86.    Solar  Eclipse. 

an  observer  at  B  will  see  the 

moon  in  the  direction  BM,  instead  of  AM.  This  will  pro- 
ject it  in  the  sky  for  the  two  observers  at  points  whose 
angular  distance  apart  is  equal  to  the  angle  BMC.  Now 
this  angle  may  be  as  great  as  1°,  approximately ;  so  that  the 
disks  of  the  moon  and  sun  might  easily  overlap  for  an 
observer  at  A,  but  not  for  an  observer  at  B  (cf.  Fig.  84, 
p.  300).  In  other  words,  solar  eclipses  are  by  no  means 

302 


ECLIPSES 

visible  throughout  an  entire  hemisphere  of  the  earth,  like 
lunar  eclipses.  In  fact,  the  distances  of  the  sun  and  moon 
from  the  earth  are  such  that  any  total  solar  eclipse  can 
be  seen  from  a  very  narrow  strip  of  the  earth's  surface 
only :  not  more  than  70  miles  wide  at  the  most,  and  ex- 
tending through  a  distance  of  much  less  than  a  hemisphere. 
Plate  20  (p.  295)  is  a  photograph  of  a  total  solar  eclipse. 

Having  thus  outlined  the  general  nature  of  eclipses  and 
their  causes,  we  shall  next  describe  certain  special  phenomena 
which  are  of  sufficient  interest  to  be  mentioned.  The  earth's 
shadow,  into  which  the  moon  enters  when  eclipsed,  is  not 
uniformly  dark  throughout.  It  is,  in  fact,  made  up  of  two 
parts, — the ' '  umbra/ '  or  shadow  proper,  and  the ' '  penumbra, ' ' 
or  partial  shadow.  Figure  87  shows  how  the  penumbra  is 
formed.  There  are  two  penumbral  regions,  as  it  were,  and 
one  umbral  region.  The  latter  is  a  central  cone  which  re- 


FIG.  87.    The  Penumbra. 

ceives  light  from  no  part  of  the  sun.  The  two  penumbral 
regions  receive  light  from  part  of  the  sun,  while  the  rest  of 
space  behind  the  earth  receives  light  from  the  entire  solar 
surface.  It  is  evident  that  the  darkness  of  the  penumbra 
will  increase  gradually  from  its  outer  edges  to  the  boundary 
lines,  where  it  gives  the  black  umbra.  The  moon,  when 
about  to  be  eclipsed,  will  therefore  enter  the  penumbra 

303 


ASTRONOMY 

first,  and  be  partially  darkened ;  and  the  darkening  will 
increase  gradually  until  it  becomes  practically  complete,  as 
the  moon  enters  the  umbra.  The  same  gradual  phenomena 
will  be  repeated  in  the  inverse  order  towards  the  end  of  the 
eclipse. 

In  the  case  of  solar  eclipses  (Fig.  88),  if  we  consider  the 
interposed  moon  as  casting  a  shadow  on  the  earth,  the  eclipse 
will  be  total  where  the  true  shadow  cone  cuts  the  earth,  and 
partial  where  the  penumbral  regions  meet  the  terrestrial 
surface. 

Owing  to  the  ellipticity  of  the  moon's  orbit,  and  consequent 
variation  in  the  distance  between  the  earth  and  moon,  it 

sometimes  happens  that  the 
true  shadow  cone  of  Fig.  88 
does  not  quite  reach  the 

FIG.  88.    Solar  Eclipse.  earth*       In    SUch    a    Case,    Ul 

that  part  of  the  terrestrial 

surface  for  which  the  eclipse  is  central,  the  sun  will  appear 
as  a  luminous  "annulus,"  or  ring,  with  the  central  part  dark. 
Such  eclipses  are  called  Annular  eclipses,  and  occur,  of 
course,  for  the  sun  only.  A  total  solar  eclipse  can  never  last 
longer  than  eight  minutes  at  any  one  place  on  the  earth, 
but  totality  in  the  case  of  the  moon  may  last  a  couple  of 
hours. 

There  exists  a  peculiar  periodicity  in  the  recurrence  of 
eclipses  called  the  Saros.  It  was  discovered  by  the  Chal- 
dseans,  who  found,  by  actual  observation,  and  comparison 
with  ancient  records  in  their  possession,  that  after  the  lapse 
of  a  period  of  6585  days  after  an  eclipse,  the  phenomenon 
will  be  repeated ;  and  eclipse  occurrences  can  thus  be  pre- 
dicted easily.  The  explanation  is  as  follows : 

The  reader  will  remember  that  in  the  case  of  the  earth's 

304 


ECLIPSES 

motion  around  the  sun  we  found  the  tropical  year  to  be 
shorter  than  the  sidereal  year  by  about  twenty  minutes, 
on  account  of  precession  of  the  equinoxes  (p.  126). 

In  a  similar  way,  on  account  of  the  motion  of  the  moon's 
nodes,  the  time  required  by  the  moon  to  travel  in  its  orbit 
from  one  node  back  to  the  same  node  again  is  shorter  than 
the  lunar  sidereal  period  (p.  161).  This  nodal  period  is 
called  the  Draconitic  period ;  it  is  three  hours  shorter  than 
the  sidereal  period,  and  two  days  seven  hours  shorter  than 
the  synodic  period.  So  we  have  : 1 

Sidereal  period       =  27d     8h 


Draconitic  period  =  27      5 


approximately 


Synodic  period       =29    12 

An  inspection  of  these  figures  shows  that  223  synodic 
periods  equal  242  draconitic  periods,  very  nearly ;  and 
either  includes  6585  days.  But  successive  full-moons,  or 
successive  new-moons,  follow  each  other  at  intervals  of 
one  synodic  month,  because  the  synodic  month  is  the  in- 
terval between  two  successive  overtakings  of  the  sun  by  the 
moon,  in  their  respective  apparent  motions  around  the 
celestial  sphere.  And  successive  passages  through  either 
node  succeed  each  other  at  intervals  of  one  draconitic  period. 
Therefore,  any  period  of  days  like  the  Saros,  containing 
exactly  a  definite  number  of  synodic  periods  and  also  a 
definite  number  of  draconitic  periods,  —  after  the  lapse  of 
such  a  period  of  days,  both  the  lunar  phase  and  the  node 
passage  must  both  repeat.  Therefore,  if  there  was  an 
eclipse  at  the  first  new  or  full  moon  of  the  Saros  period,  there 
must  also  be  an  eclipse  at  the  first  new  or  full  moon  of  the 
succeeding  Saros  period. 

1  Note  37,  Appendix. 
x  305 


ASTRONOMY 

In  the  light  of  the  above  explanation  of  eclipses,  it  may 
be  possible  to  make  somewhat  clearer  the  allied  phenome- 
non called  a  transit  of  Venus  (p.  268).  Such  transits  also 
occur  in  the  case  of  Mercury,  but  they  are  then  of  lesser  in- 
terest. The  planetary  nodes  do  not  move  around  the 
ecliptic  rapidly,  like  the  lunar  nodes;  they  remain  almost 
stationary  at  a  definite  point.  The  sun,  in  its  apparent 
motion  around  the  ecliptic,  reaches  the  nodal  points  of 
Venus  on  June  5  and  December  7 ;  so  that  transits  of  that 
planet  happen  (if  at  all)  within  a  day  or  two  of  these  dates. 
To  ascertain  the  interval  between  successive  transits,  we 
note  that : 

5  synodic  periods  of  Venus  =      8  years,  nearly  ; 
152  synodic  periods  of  Venus  =  243  years,  very  nearly. 

Since  conjunctions  of  Venus  occur  at  intervals  of  one  sy- 
nodic period,  any  given  transit  may  be  followed  by  another 
at  the  same  node  eight  years  later.  But  there  could  not  be 
a  third  transit  sixteen  years  later;  the  eight-year  period  is 
not  exact  enough  for  that.  We  should  then  have  to  wait 
for  the  243-year  period  to  become  effective.  But  at  the  other 
node,  a  transit,  or  an  eight-year  pair  of  transits,  may  happen 
after  half  the  243-year  cycle  has  elapsed. 


306 


CHAPTER  XVIII 

COMETS 

THE  comets,  stellce  cometce,  or  stars  with  hair,  must  next 
receive  our  attention.  These  bodies  usually  move  in  very 
elongated  orbits,  with  the  sun  at  one  focus.  They  often 
come  as  mere  occasional  visitors  to  the  solar  system,  are 
seen  during  a  short  period  only,  while  they  are  traversing 
that  part  of  their  orbit  which  is  near  the  sun  and  therefore 
also  near  the  terrestrial  orbit.  Occasionally  comets  have 
been  as  luminous  as  the  brightest  planet  (Venus) ;  have 
sometimes  been  seen  in  daylight;  and  very  often  have 
a  long  appendage  streaming  out  from  the  head,  —  the 
comet's  "tail." 

It  is  easy  to  enumerate  the  chief  known  facts  concerning 
the  comet's  physical  appearance.  The  head  usually  con- 
sists of  a  "coma,"  or  hazy  nebula,  containing  a  "nucleus," 
or  central  condensation.  Attached  to  it  is  the  tail,  or,  as 
it  was  sometimes  formerly  called,  the  "beard."  The  coma 
is  the  part  that  generally  becomes  visible  first,  as  the  body 
begins  to  approach  the  solar  system.  The  nucleus  forms 
later,  or  at  least  becomes  visible  later.  The  tail,  strange 
to  say,  is  always  directed  away  from  the  sun;  so  that 
when  the  comet  is  receding  from  the  sun,  after  passing  the 
perihelion  point  (p.  120)  of  its  orbit,  it  pushes  its  tail  out 
ahead  of  it.  But  many  comets  were  not  discovered  until 
after  perihelion.  It  was  then  that  the  astronomers  of  old 
used  the  name  "beard"  for  the  tail. 

Comets  are  big;  their  volume  is  sometimes  incredibly 

307 


ASTRONOMY 

large.  The  heads  run  up  to  a  million  miles  in  diameter, 
and  the  tails  may  be  ten  million  miles  long,  or  even  much 
longer.  But  they  have  little  mass,  as  is  evidenced  by  the 
total  absence  of  gravitational  perturbations  (p.  206)  in  the 
motions  of  the  earth  and  Venus,  even  when  a  big  comet 
passes  very  near  these  planets.  Owing  to  this  vast  ness  of 
bulk  and  extremely  small  mass,  comets  have,  of  course, 
a  very  low  density,  especially  in  their  tails.  Moreover, 
stars  have  at  times  been  seen  through  the  comets,  even 
through  their  heads. 

In  view  of  this  extreme  lack  of  mass,  it  may  seem  strange 
that  modern  science  is  compelled  to  admit  the  possibility, 
at  least,  of  danger  resulting  from  collision  between  the 
earth  and  a  comet.  If  the  cometary  particles  are  infinitesi- 
mally  small,  no  injury  would  follow ;  but  if  the  particles  are 
rocks  weighing  tons,  they  might  cause  considerable  local 
damage  at  the  point  of  collision  on  the  earth. 

But,  on  the  whole,  the  most  plausible  theory  is  to  suppose 
these  bodies  to  be  composed  of  tiny  particles  traveling 
together  in  swarms,  and  separated  by  distances  many 
times  greater  than  the  diameter  of  the  particles.  And  the 
particles  may  be  surrounded  by  atmospheres  of  incan- 
descent gases ;  for  we  know  that  comets  are  partly  self- 
luminous,  although  they  send  us  also  a  certain  amount  of 
reflected  sunlight,  like  the  planets.  This  is  made  certain 
by  observations  of  their  spectra,  which  generally  show  the 
existence  of  hydrocarbon  gas  in  a  luminous  state,  as  well  as 
a  dim  continuous  spectrum  containing  Fraunhofer  (p.  287) 
lines,  —  the  sure  indication  of  solar  light. 

We  have  a  good  theory  to  account  for  the  repulsive  forces 
that  must  come  from  the  sun,  so  as  to  make  the  cometary  tail 
always  point  away  from  that  body.  The  researches  of 

308 


COMETS 

Clerk-Maxwell  (the  same  who  proved  mathematically 
the  satellite  construction  of  Saturn's  ring,  p.  245)  have 
brought  out  the  fact  that  light-rays  exert  a  slight  physical 
pressure  upon  any  object  they  reach.  This  pressure  is 
theoretically  proportional  to  the  area  illuminated. 

Now  if  we  imagine  a  small  spherical  cometary  particle, 
and  suppose  its  radius  to  diminish,  the  light  pressure  will 
diminish  in  proportion  to  the  square  of  the  radius,  because 
the  area  of  the  circular  cross-section  illuminated  diminishes 
in  that  proportion.  But  the  volume  and  mass  of  the  particle 
will  diminish  as  the  cube  of  the  radius.  Therefore  the  mass 
diminishes  more  rapidly  than  the  area.  But  the  solar  gravi- 
tational attraction  is  proportional  to  the  mass ;  consequently, 
the  solar  attraction  diminishes  more  rapidly  than  the  light 
pressure,  as  the  particle  grows  smaller.  We  have  therefore 
merely  to  imagine  the  particle  small  enough,  and  the  light 
pressure  will  balance  the  attraction.  Still  smaller  particles 
will  actually  be  repelled. 

If  we  now  suppose  the  tail  composed  of  particles  smaller 
than  those  in  the  head,  everything  is  explained :  the  head 
attracted,  the  tail  repelled,  by  the  sun.  Probably  the 
comet  has  no  tail  until  it  approaches  the  sun,  when  the  light 
pressure  sifts  out  the  small  particles ;  repells  them  in  an 
increasing  degree  as  the  comet  comes  near  perihelion ;  and 
thus  makes  the  tail  "grow."  Perhaps  the  tail  particles 
never  rejoin  the  head,  but  are  left  scattered  throughout  the 
length  of  the  cometary  orbit.  If  we  finally  suppose  both 
gravitational  attraction  and  light  repulsion  to  be  exerted 
on  the  several  particles,  both  by  the  sun  and  the  comet's 
head,  we  have  a  combination  of  forces  sufficiently  flexible 
to  account  for  the  most  complicated  observed  forms  of 
cometary  tails.  (See  Plate  21,  p.  307.) 

309 


ASTRONOMY 

The  number  of  comets  is  very  great ;  while  only  some  four 
hundred  were  observed  before  1610,  when  Galileo  first 
used  the  telescope,  at  least  four  hundred  more  have  been 
found  in  the  succeeding  three  centuries.  Of  course  this 
greater  abundance  of  discovery  in  modern  times  has  been 
brought  about  by  the  possibility  of  recording  comets  too 
faint  to  be  seen  by  the  unaided  eye.  Only  thirteen  naked- 
eye  comets  belong  to  the  nineteenth  century. 

The  method  used  in  discovering  comets  is  interesting. 
This  work  is  carried  on  by  specialists ;  except  as  a  result  of 
unusual  chance,  one  can  expect  to  find  new  comets  only  after 
a  number  of  years'  severe  study  of  the  heavens.  The  usual 
process  is  to  "sweep"  the  sky  with  a  telescope  of  moderate 
size  and  low  magnifying  power.  Any  hazy  object  may 
be  a  comet ;  for  when  they  are  distant  and  dim,  comets 
always  look  more  or  less  like  small  nebulae.  The  only 
sure  test  to  distinguish  them  is  to  watch  for  an  hour  or 
two,  and  ascertain  whether  there  is  motion  relative  to  the 
fixed  stars.  If  there  is  motion,  a  comet  has  been  found. 
But  this  motion  test  occupies  much  time ;  and  at  this  point 
the  experience  of  years  is  of  value.  For  the  comet  hunter 
learns  at  last  to  know  all  the  tiny  configurations  of  stars 
seen  in  the  telescope ;  he  knows  the  telescopic  constellations 
at  sight,  as  well  as  most  astronomers  know  Orion  and  the 
Great  Bear.  It  is  said  that  Olbers,  for  instance,  could  tell 
the  approximate  right-ascension  and  declination  of  the 
point  on  the  sky  toward  which  his  telescope  was  directed, 
by  simply  looking  through  the  eye-piece,  and  noting  the 
diagram  of  stars  appearing  in  the  field  of  view. 

And  the  comet  hunter  must  know  all  the  little  nebulse  too, 
as  well  as  their  positions  on  the  sky  relative  to  the  sur- 
rounding small  stars.  When  he  sweeps  a  faint  nebulous  ob- 

310 


PLATE  22.     Halley's  Comet. 


Photo  by  Curtis. 


COMETS 

ject  into  view,  he  does  not  ordinarily  need  to  delay  his  work 
by  testing  for  motion;  he  recognizes  the  object  at  once,  if 
it  is  one  of  the  known  small  nebulae. 

Comets  are  usually  named  after  their  discoverer,  though 
a  few  bear  the  name  of  some  person  who  has  explained  their 
motions  or  peculiarities  by  a  special  investigation.  Thus 
the  famous  comet  of  Halley,  a  photograph  of  which  is  shown 
in  the  accompanying  Plate  22,  was  the  first  comet  for  which 
future  returns  to  the  solar  system  were  predicted  by  means 
of  orbital  calculations.  These  were  made  by  Halley  (cf. 
p.  269) ;  and  his  name  was  accordingly  assigned  to  this  comet. 

Small  telescopic  comets  are  at  first  designated  by  the 
year  of  discovery  and  a  letter ;  as,  1899  a,  etc.  Later,  when 
orbits  have  been  computed,  they  take  the  number  of  the 
year  in  which  their  closest  approach  to  the  sun,  or  perihelion, 
occurs;  and  a  number  in  addition  to  show  the  order  of 
cometary  perihelion  passages  during  that  year.  Thus 
Donati's  great  comet  of  1858  was  1858  /,  or  the  sixth  comet 
discovered  in  1858;  and  later  it  became  1858  VI,  or  the 
one  whose  perihelion  passage  was  the  sixth  perihelion  passage 
in  1858. 

The  period  of  visibility  does  not  usually  last  more  than  a. 
few  months,  although  its  average  duration  has  been  length- 
ened considerably  in  recent  years,  because  modern  giant  tele- 
scopes can  observe  the  comets  long  after  their  orbital  motion 
has  carried  them  quite  beyond  the  range  of  ordinary  glasses. 

Concerning  these  orbital  motions,  the  mists  .of  antiquity 
certainly  enshroud  some  very  singular  notions.  There  was 
a  time  when  comets  were  regarded  as  material  thrown  out 
from  the  earth,  possibly  through  volcanoes.  It  was  not 
until  1577  that  Tycho  Brahe  for  the  first  time  proved  from 
actual  measurements  that  the  great  comet  of  that  year  was 

311 


ASTRONOMY 

surely  farther  from  the  earth  than  is  the  moon.  Kepler 
thought  comets  are  alive.  Hooke,  in  1675,  a  century  after 
Tycho  Brahe,  suggested  that  comet  orbits  might  be  para- 
bolic :  a  very  few  years  later,  Newton  showed  that  they 
are  " conic  sections,"  and  Halley  calculated  actual  orbits 
for  all  the  comets  observed  up  to  that  time. 

There  are  three  kinds  of  conic  sections, — the  ellipse,  para- 
bola, and  hyperbola ;  and  it  is  easy  to  draw  a  figure  illustrat- 
ing these  three  curves  as  comet  orbits.  Parabolic  orbits 
are  four  times  as  frequent  as  elliptic  orbits  :  hyperbolas  are 


FIG.  89.     Forms  of  Comet  Orbits. 

very  rare,  and  it  is  not  absolutely  certain  that  any  such  orbits 
really  exist.  Figure  89  exhibits  the  three  orbital  forms, 
together  with  the  comparatively  tiny  circular  terrestrial 
orbit,  and  the  sun-dot  at  its  center.  We  see  especially 
how  nearly  alike  all  three  kinds  of  comet  orbits  are,  while 

312 


COMETS 

the  comet  is  near  the  earth's  path;  and,  after  all,  it  is  only 

then  that  we  can  see  a  comet,  and  observe  its  position, 

as  projected  on  the  sky.     To  construct  a  comet  orbit  from 

observations  is  often  as  difficult  as  trying  to  draw  a  circle 

of  large  radius  through  three  points  very  close  together. 

Thus,  in  Fig.  90,  it  is 

easy  to  draw  the  circle 

A  accurately,  through 

the  three  points  PI,  P2, 

P3.      But  if  we  were 

asked  to  draw  a  circle 

through  P/,  P2',  P8', 

we  might  not  be  able 

to  decide  between  the  circle  B  and  the  circle  C. 

Now  the  ellipse  is  a  closed  curve ;  the  others  are  open 
curves :  therefore  only  elliptic  orbits  will  produce  so-called 
"periodic"  comets,  with  future  returns  to  the  solar  system. 
The  parabolic  comets  visit  us  once,  and  never  return.  But 
since  parabolic  orbits  closely  resemble  extremely  elongated 
ellipses,  it  is  not  always  possible  to  make  certain  whether 
any  given  object  is  periodic  or  not.  But  it  is,  after  all,  really 
immaterial  whether  a  given  comet  orbit  is  truly  parabolic,  or 
elliptic,  with  a  period  of  several  hundred  thousand  years. 

The  exact  details  of  any  comet  orbit  are  defined  by  means 
of  elements  exactly  analogous  to  the  elements  of  a  planet's 
orbit  (p.  200). 

So  much  being  premised  about  these  orbits,  we  can  now 
consider  one  of  the  most  interesting  things  known  about 
comets, — their  " families."  For  there  are  in  existence  most 
curious  kinships  between  various  groups  of  comets.  Coming 
back  to  Kepler's  amusing  notion  that  they  are  alive,  we 
must  expect  to  find  close  relationships  among  them,  and 

313 


ASTRONOMY 

also  some  that  are  merely  distant  cousins,  as  it  were.  The 
most  remarkable  "  blood-relations "  are  the  great  comets  of 
1668,  1843,  1882,  and  1887.  They  must  be  brother-comets, 
for  they  all  pursue  practically  the  same  orbit,  though  travel- 
ing in  different  parts  of  it.  They  approached  the  solar 
system  from  the  direction  of  the  bright  star  Sirius,  and  left 
again  in  nearly  the  same  direction,  in  a  parabolic  orbit. 

On  the  other  hand,  there  are  the  comet-families  belonging 
to  the  great  planets,  especially  Jupiter.  Here  all  the 
comets  of  a  family  have  the  peculiarity  that  the  points  of 
their  orbits  farthest  from  the  sun,  the  aphelion  points, 
all  lie  near  the  orbit  of  Jupiter.  In  other  words,  they  recede 
from  the  sun  just  far  enough  to  reach  Jupiter's  orbit.  If 
Jupiter  happens  to  be  in  the  neighborhood  when  they  get 
out  there,  he  must  exert  a  powerful  gravitational  attraction 
upon  them.  It  is  supposed  that  on  their  first  arrival,  per- 
haps in  parabolic  orbits,  this  attraction  pulled  them  around 
into  ellipses,  having  their  node  and  aphelion  point  near  the 
place  where  this  disturbing  pull  took  place.  This  is  the  well- 
known  capture  theory  of  comets,  due  to  Laplace. 

So  we  see  that  the  comets  did  not  originally  belong  to  our 
solar  system;  they  come  to  us  from  outer  space,  possibly 
from  among  the  fixed  stars,  possibly  from  some  nearer  region. 
If  they  come  from  interstellar  spaces,  we  should,  on  the 
whole,  expect  to  find  a  preponderance  of  orbits  having  their 
aphelion  points  lying  in  the  direction  of  that  point  on  the 
sky  toward  which  the  solar  system's  own  motion  in  space  is 
tending.  For  the  solar  system,  as  a  whole,  is  drifting  through 
cosmic  space,  as  will  be  explained  in  a  later  chapter.  But 
we  have  only  slight  indications  of  such  a  clustering  of  aphelion 
points :  our  whole  theory  as  to  comet  origins  is  till  hazy, 
very  hazy. 

314 


CHAPTER  XIX 

METEORS   AND   AEROLITES 

THE  consideration  of  comets  leads  us  directly  to  the 
closely  related  sub j  ect  of  meteors  or ' '  shooting  stars . ' '  These 
look  like  stars  falling  from  the  sky ;  actually,  they  are  small 
particles  of  matter  traveling  in  space,  and  passing  through  the 
earth's  atmosphere.  They  give  a  bright  light,  and  usually 
leave  a  long  visible  trail  behind  them.  Sometimes  they  do 
not  appear  merely  as  isolated  bodies;  but  regular  showers 
occur,  with  the  bright  intermittent  trails  almost  covering 
the  sky,  or  a  portion  of  it,  for  a  con- 
siderable time.  When  this  happens,  it 
has  been  found  that  all  the  meteor 
trails  are  directed  away  from  some 
single  point  on  the  sky.  This  point 
is  called  the  Radiant ;  and  it  has  the 
peculiarity  that  the  trails  are  always 
short  in  its  vicinity.  Figure  91  exhib- 
its this  state  of  affairs.  The  point  R 
on  the  celestial  sphere  is  the  radiant.  FIG.  91.  Radiant  of  Meteor 

Shower. 

All  the  trails  are  directed  away  from 

it,  as  shown  by  the  arrows ;  and  the  longer  trails  originate 

at  points  farther  from  the  radiant  than  do  the  short  trails. 

Figure  92  shows  that  the  whole  appearance  is  due  simply 
to  perspective.  The  meteors  move  in  parallel  lines  to  meet 
the  earth.  Suppose  the  observer  to  be  on  the  surface  of 
the  earth,  at  0,  and  two  meteors  moving  along  parallel  lines, 

315 


ASTRONOMY 

such  as  AiBi  and  A^B^.  To  the  observer  at  0  the  meteors 
will  seem  to  move  along  the  lines  A\C\  and  A2(72.  And  all 
the  meteors  will  seem  to  move  along  lines  that  will  appear 
to  radiate  out  from  a  single  point  R,  where  AiCi  and  A2C2 
intersect,  if  produced  backwards  from  Ai  and  A2.  And 

this  point  will  appear 
in  the  sky  near  the 
meteors  that  seem  to 
have  short  trails. 

Each  meteor  shower 
can    be    distinguished 
s^ce  of  the  Earth  ~  frOm  all  others  ;  not  by 

FIG.  92.    Radiant  of  Meteor  Shower.  , .  „,,  .        , 

a  difference  in  the  ap- 
pearance of  its  constituent  meteors,  but  by  the  position  on 
the  sky  of  its  radiant.  Thus  the  shower  called  the  Leonids, 
the  greatest  of  all  the  showers,  has  its  radiant  in  the  con- 
stellation Leo  (Fig.  20,  p.  62).  These  meteors  occur  always 
about  November  12,  and  have  been  found  especially  numerous 
at  intervals  of  33  years. 

The  cause  of  this  fixity  in  the  dates  and  recurrences  of 
individual  showers  is  quite  simple.  Each  shower  travels 
in  a  definite  orbit  around  the  sun,  just  like  a  periodic  comet 
(p.  313).  This  orbit  somewhere  intersects  the  orbit  of  the 
earth;  or,  at  least,  passes  very  near  it.  But  the  earth 
must  reach  that  point  of  intersection  on  the  same  date  each 
year.  Therefore  the  shower  must  occur  on  that  particular 
date,  if  it  occurs  at  all.  And  it  will  occur  if  the  meteors  hap- 
pen to  be  at  the  intersection  point  of  the  two  orbits  on  the 
date  when  the  earth  also  reaches  that  point.  In  the  case  of 
the  November  Leonids  this  happens  only  once  in  33  years ; 
but  the  Perseids,  or  August  meteors,  are  ready  for  us  every 
year.  We  conclude,  of  course,  that  the  Perseids  are  spread 

316 


METEORS   AND   AEROLITES 


C'irc/e 


out  all  along  their  orbit,  so  that  we  meet  some  of  them  when- 
ever we  strike  the  orbit.  But  the  Leonids  must  be  concen- 
trated in  a  certain  region  in  their  orbit :  this  region  comes 
around  to  the  point  of  intersection  in  the  proper  way  only 
once  in  33  years. 

A  very  interesting  fact  about  the  meteors  is  that  we  or- 
dinarily observe  more  of  them  per  hour  just  before  sunrise 
than  we  do  just  after 
sunset.  The  reason  is 
shown  in  Fig.  93.  EI  is 
the  position  of  the  earth 
in  its  orbit  at  about  six 
in  the  morning,  just  be- 
fore sunrise.  The  sun  is 
seen  projected  on  the 
ecliptic  at  /Si,  which  is 
therefore  near  the  east 
point  of  the  horizon. 
The  earth's  orbital  mo- 
tion is  for  the  moment 
directed  towards  the 
point  P,  90°  west  of  S, 
where  the  sun  appears  on  the  ecliptic.  The  earth's  orbital 
motion  takes  place  in  the  direction  of  the  curved  arrow,  so 
that  at  six  in  the  morning  we  are  on  the  front  of  the  earth 
in  respect  to  its  orbital  motion;  we  advance  to  meet  the 
meteors.  But  on  the  opposite  side  of  the  earth  it  is  at  the 
same  instant  6  o'clock  in  the  evening.  There  they  see  only 
such  meteors  as  overtake  the  earth,  while  on  the  front  side 
we  see  them  all.  After  the  lapse  of  twelve  hours,  the  earth 
has  made  a  half-turn  on  its  rotation  axis ;  conditions  are 
reversed  ;  and  we  then  have  our  clocks  at  six  in  the  evening. 

317 


FIG.  93. 


90°  West  off  he  Sun 
Meteors  at  Sunrise  and  Sunset. 


ASTRONOMY 

We  are  then,  in  our  turn,  on  the  back  of  the  earth  with 
respect  to  the  direction  of  its  orbital  motion. 

Meteors  are  never  seen  until  they  enter  the  atmosphere  of 
our  earth,  but  their  heat  and  light  are  not  due  to  atmos- 
pheric friction  in  the  ordinary  sense.  Sometimes  it  is  said, 
erroneously,  that  they  are  "set  on  fire"  in  the  same  way  as 
a  friction  match  is  lighted  by  being  rubbed  on  a  rough 
surface.  Their  light  is  really  caused  by  the  compression  of 
air  in  advance  of  the  moving  meteor.  It  is  harder  for  the 
meteor  to  move  against  the  compressed  air ;  this  retards  its 
motion,  and  the  motion  energy  is  transformed  into  heat  energy 
(p.  2).  Doubtless  both  the  meteor  and  the  air  are  heated* 

When  a  moving  body  is  retarded  by  the  resistance  of 
an  atmosphere,  the  heat  engendered  is  proportional  to  the 
square  of  the  velocity  of  motion.  At  the  usual  meteoric  ve- 
locity, the  temperature  produced  is  probably  equivalent  to 
several  thousand  degrees  Fahrenheit;  and  this,  of  course, 
will  melt  almost  any  substance.  The  rapid  motion  through 
the  air  then  tears  off  particles  of  heated  incandescent  matter 
from  the  melted  meteoric  surface ;  and  these  particles  are  left 
behind  to  form  the  tail  or  trail  of  the  meteor.  It  is  not 
known  just  why  it  sometimes  remains  visible  for  many 
minutes.  Plate  23  is  a  photographic  reproduction  of  a 
meteor  trail,  showing  two  remarkable  variations  of  brilliancy. 
The  negative  also  contains  a  couple  of  interesting  nebulae  of 
irregular  form. 

It  is  altogether  probable  that  the  meteors,  and  especially 
the  meteoric  showers,  are  nothing  else  but  fragments  of  dis- 
integrated comets.  As  soon  as  periodicity  in  the  recurrence 
of  showers  was  recognized,  and  it  thus  became  plain  that  the 
meteors  travel  in  definite  orbits,  it  was  but  a  short  step  to 
compare  those  orbits  with  known  cometary  paths.  And 

318 


PLATE  23.     Meteor  Trail. 


Photo  by  Barnard. 


METEORS  AND   AEROLITES 

soon  after  the  great  Leonid  shower  of  1866,  Schiaparelli 
showed  that  the  Perseids,  or  August  meteors,  are  in  the  same 
orbit  as  a  comet  discovered  by  Tuttle  in  1862.  And  it 
was  not  long  before  the  Leonids  were  similarly  identified 
with  the  comet  of  1866,  discovered  by  Temple. 

At  least  eight  different  meteor  showers  are  now  known  to 
be  connected  with  comets.  The  conclusion  is  possible  that 
the  comet  is  itself  but  a  condensed  place  in  the  meteoric 
procession;  the  meteors  themselves  the  disintegrated  part 
of  the  material  involved  in  the  whole  transaction.  Certain 
it  is  that  at  least  one  comet  (Biela's)  has  actually  been  seen 
to  break  up.  It  was  discovered  in  1826,  re-appeared  in 
1846  according  to  prediction,  and  was  seen  to  break  in  two 
during  this  period  of  visibility.  In  1852  it  was  seen  again, 
the  two  parts  being  now  widely  separated ;  and  it  has  never 
been  visible  since.  In  its  orbit,  however,  moves  one  of  the 
big  meteor  swarms. 

It  is  very  important  to  determine  as  accurately  as  possible 
the  height  of  meteors  above  the  earth's  surface.  For  this 
is  about  the  only  direct 
method  we  have  to  ascer- 
tain observationally  the 
extent  of  the  terrestrial 
atmosphere.  Since  the 
meteor  becomes  visible 
only  when  it  penetrates 

,       '  , ,  .  FIG.  94.     Height  of  Meteors. 

the    earth  s    envelope    of 

air,  we  shall  know  something  about  the  height  of  the  air 
if  we  can  measure  the  height  of  the  meteor.  The  only  way 
to  do  this  is  to  select  a  couple  of  stations  on  the  earth 
and  make  simultaneous  observations  of  the  same  meteor. 
Figure  94  shows  how  this  is  done. 

319 


ASTRONOMY 

If  observers  on  the  earth's  surface,  at  Oi  and  02,  see  the 
meteor  M  projected  on  the  celestial  sphere  near  the  stars  Si 
and  $2,  it  is  clear  that  we  can  calculate  the  height  MH 
of  the  meteor  from  the  known  distance  0i02  between  the 
observers,  and  the  angles  $iOi02  and  S2020i,  which  are,  of 
course,  known,  if  we  know  the  positions  of  the  stars  Si  and 
S%  on  the  sky. 

But  there  is  great  difficulty  in  securing  these  observations 
in  such  a  manner  as  to  be  certain  that  they  apply  to  the 
same  meteor.  It  is  necessary  to  make  the  attempt  on 
some  night  when  numerous  meteors  are  to  be  expected ; 
and  it  is  then  by  no  means  easy  to  be  certain  that  the  ob- 
servations of  M  from  the  two  stations  are  really  simultaneous, 
and  apply  to  but  a  single  meteoric  object.  Such  as  they  are, 
observations  of  this  kind  indicate  an  extreme  height  of 
75  miles  for  our  atmosphere. 

Having  thus  explained  the  principal  facts  about  the 
meteors,  we  come  next  to  the  Aerolites.  These  are  stones, 
or  pieces  of  iron  mixed  with  other  materials,  which  fall  upon 
the  surface  of  the  earth  from  time  to  time.  There  is  little 
doubt  of  their  being  meteors  that  actually  strike  the  earth, 
probably  on  account  of  unusually  large  size.  For  a  very 
big  meteor  would  not  be  entirely  consumed,  as  it  were,  in  its 
passage  through  the  air,  and  might  be  attracted  down  to 
the  surface  of  the  earth  by  the  gravitational  pull  of  the  planet. 
Those  that  are  completely  consumed  perhaps  fall  on  the 
earth  finally  as  dust :  certain  it  is  that  dust,  probably 
meteoric,  has  been  found  on  the  surface  of  ancient  Arctic 
ice.  This  is  the  so-called  "cosmic  dust."  Many  specimens 
of  aerolites  are  preserved  in  museums.  A  number  have 
actually  been  seen  to  fall,  so  there  is  no  doubt  whatever  as 
to  their  origin  being  outside  the  earth  itself.  When  seen 

320 


METEORS   AND   AEROLITES 

at  night  they  exhibit  a  bright  round  head,  with  a  luminous 
trail.     Occasionally  there  is  an  audible  explosion. 

The  outer  surf  aces  of  the  aerolitic  specimens  in  our  museums 
seem  to  have  been  melted,  showing  the  effect  of  the  high 
temperatures  produced  through  their  impeded  motion  in 
the  air.  Chemically,  they  contain  only  elements  or  sub- 
stances known  on  the  earth. 


321 


CHAPTER  XX 

STARSHINE 

IN  the  preceding  chapters  we  have  completed  a  somewhat 
detailed  description  of  the  solar  system,  and  are  now  ready 
to  proceed  outward  into  space,  to  study  the  distant  uni- 
verse of  stars.  Astronomers  believe  that  an  analogy  exists, 
more  or  less  close,  between  our  sun  and  the  stars  (p.  6). 
Together  with  its  system  of  planets,  the  sun  may  be  regarded 
as  a  small  isolated  group  suspended  in  space,  and  separated 
from  other  similar  groups  by  distances  almost  incomparably 
greater  than  any  existing  within  the  solar  system  itself. 
We  may  be  quite  sure  that  the  stars  are  all  excessively  dis- 
tant :  we  are  troubled  by  no  doubts  in  this  respect,  as  were 
the  ancients.  For  we  now  have  the  law  of  gravitation; 
from  it  we  know  that  if  there  were  a  celestial  body  of  any 
kind  in  space,  as  massive  as  the  sun,  and  not  more  than  ten 
thousand  times  as  far  away  as  the  distance  separating  the 
earth  from  the  sun,  —  that  body  would  surely  reveal  its 
existence  through  observable  perturbations  (p.  206)  pro- 
duced in  the  motions  within  our  solar  system.  And  this  it 
would  do,  even  if  it  were  a  dead  sun,  no  longer  luminous, 
and  quite  invisible.  To  produce  gravitational  attraction, 
and  consequent  perturbative  effects,  merely  the  presence  of 
matter  would  be  necessary,  not  visible  matter. 

Furthermore,  actual  measurements,  to  be  described 
later,  have  shown  that  the  nearest  fixed  star  so  far  observed 
is  more  than  200,000  times  as  far  from  the  sun  as  is  the  earth. 

322 


PLATE  24.     The  Constellation  Serpentarius. 
(From  Hevelius'  Prodromus  Astronomiae,  Gedani,  1690.) 


STARSHINE 

But  this  last  argument  is  not  conclusive,  because  we  can 
measure  only  visible  stars,  and  the  nearest  one  might  con- 
ceivably be  non-luminous.  This  objection,  of  course, 
does  not  apply  to  the  gravitational  argument. 

We  have  seen  (p.  6)  that  the  stars  are  classified  according 
to  their  magnitudes,  and  that  this  term  " magnitude" 
does  not  here  have  its  usual  meaning.  It  has  no  relation  to 
size  or  bigness,  but  simply  indicates  the  degree  of  luminosity 
or  brightness  of  a  star.  We  shall  now  consider  this  matter 
somewhat  more  in  detail.  Old  Hipparchus  (pp.  127,  189, 
297)  was  the  first  to  divide  the  stars  into  magnitude  classes ; 
he  simply  selected  arbitrarily  the  twenty  brightest  stars 
he  could  see,  and  called  them  first-magnitude  stars.  He 
then  designated  as  sixth-magnitude  all  objects  that  belonged 
at  his  absolute  lower  limit  of  vision,  —  that  he  could  just 
see,  though  with  difficulty.  Stars  of  intermediate  luminosity 
he  placed  in  intermediate  classes,,  also  somewhat  arbitrarily. 
This  gave  a  rather  rough  classification ;  but  it  is  still  in  use 
(with  some  improvements)  down  to  the  present  day. 

Adopting  Hipparchus'  magnitude  scale,  we  find  the  num- 
ber of  stars  of  the  various  magnitudes  situated  between  the 
north  pole  of  the  heavens  and  the  circle  of  declination  35° 
south  of  the  celestial  equator  to  be  as  follows : 

1st  mag.  14,     3d  mag.  152,     5th  mag.  854, 
2d  mag.  48,     4th  mag.  313,     6th  mag.  2010, 
Total,  3391. 

The  above  rough  system  of  classification  was  replaced  by 
a  more  exact  one  about  1850.  Sir  John  Herschel  had 
remarked  from  his  photometric  observations  that  first- 
magnitude  stars  average  just  about  100  times  the  luminosity 
of  sixth-magnitude  stars.  So  it  was  suggested  that  we  take 
the  exact  "  fifth  root  "  of  100  as  the  ratio  between  the  lumi- 

323 


ASTRONOMY 

nosities  of  any  two  successive  star-magnitudes.  And  this 
ratio  is  called  the  "light-ratio." 

In  this  way,  we  make  the  fifth-magnitude  stars  VlOO 
times  as  bright  as  the  sixth-magnitude  stars;  the  fourth 
VlOO  times  as  bright  as  the  fifth;  etc.  Consequently,  the 
first-magnitudes  would  be  VlOO  X  VlOO  X  VlOO  X  VlOO 
X  VlOO,  or  100,  times  as  bright  as  the  sixth-magnitudes, 
in  exact  accord  with  Herschel's  observation. 

Now  the  fifth  root  of  100  is  about  2|,  so  that  stars  of  any 
magnitude  are  approximately  two-and-one-half  times  as 
bright  as  those  of  the  next  fainter  magnitude.  To  fix  a 
definite  zero  for  this  scale,  it  has  been  decided  to  select 
certain  stars  as  standards.  Thus,  Aldebaran  is  a  standard 
first-magnitude  :  other  stars  can  be  compared  with  it ;  their 
light- ratio  measured  by  observation;  and  the  magnitude 
difference  then  ascertained.1  This  process,  of  course,  some- 
times assigns  zero-magnitude,  or  even  a  negative  magnitude, 
to  an  exceptionally  bright  star,  like  Sirius.  On  the  above 
scale,  Sirius  actually  comes  out  from  photometric  observa- 
tions as  minus  1.4.  The  sun's  stellar  magnitude  is  about 
—  26,  the  enormous  luminosity  being  in  this  case  due  to 
proximity,  not  to  intrinsic  light-giving  power. 

Observations  of  the  relative  luminosities  of  stars  are  made 
with  an  instrument  called  an  astronomic  photometer.  The 
ordinary  telescope  may  be  so  used  in  a  very  simple  way. 
To  estimate  a  star's  brightness,  we  have  only  to  place  dia- 
phragms pierced  with  holes  of  various  sizes  outside  the 
object-glass  (p.  272)  until  we  find  one  that  will  just  allow 
us  to  glimpse  the  star.  It  is  obvious  that  this  method  is 
possible,  for  if  we  use  successively  a  series  of  diaphragms 
with  apertures  diminishing  gradually,  we  shall  in  effect  be 

1  Note  38,  Appendix. 
324 


STARSHINE 

making  the  telescope  smaller  and  smaller,  and  there  must 
come  a  time  when  it  has  been  made  so  small  that  it  will 
just  fail  to  show  the  star  under  observation.  From  the  size 
of  the  aperture  in  this  last  diaphragm,  it  is  possible  to  cal- 
culate the  luminosity  of  the  star.1 

There  are  also  other,  and  perhaps  better,  forms  of  pho- 
tometers, in  which  the  star  under  examination  is  compared 
with  an  artificial  star  produced  by  a  light  in  the  observatory 
placed  behind  a  screen  having  a  very  small  hole.  Varying 
the  artificial  star  until  it  appears  of  the  same  luminosity 
as  the  real  one  enables  the  observer  to  measure  accurately 
the  brightness  of  the  latter.  Magnitudes  may  also  be 
measured  photographically.  The  little  dots  produced  on 
a  sensitive  plate  by  prolonged  exposure  in  the  telescope 
vary  in  a  sort  of  proportion  to  star-magnitudes :  the  bright 
stars  produce  larger  dots.  Therefore  a  microscopic  measure- 
ment of  the  dot  diameters  on  an  astronomic  negative  enables 
us  to  estimate  star-magnitudes. 

But  all  astronomic  photometric  measures  are  subject  to 
considerable  error  on  account  of  "  light-absorption "  in 
the  terrestrial  atmosphere.  Some  of  the  stellar  light  is  lost 
in  passing  through  our  air.  This  effect  is,  of  course,  smallest 
for  stars  near  the  zenith;  for  there  light  passes  straight 
through  the  atmospheric  layer,  and  at  right  angles  to  it. 
The  path  through  the  air  is  thus  the  shortest  possible.  But 
for  stars  near  the  horizon  the  light  enters  the  atmosphere  at  a 
rather  small  angle,  and  its  path  is  much  longer,  before  it 
reaches  the  observer's  eye.  Consequently,  stars  are  bright- 
est when  they  are  near  the  zenith. 

How  much  is  the  total  light  received  from  the  stars  ? 
This  question  has  been  widely  studied,  but  only  the  very 

1  Note  39,  Appendix. 
325 


ASTRONOMY 

roughest  results  have  been  obtained.  Possibly  the  whole 
sidereal  heavens  give  about  as  much  light  as  2000  stars 
like  Vega.  This  is  approximately  •£$  of  full  moonlight  ; 
and  it  includes  the  considerable  quantity  of  starlight  coming 
from  objects  below  the  sixth  magnitude,  and  therefore 
invisible  to  the  unaided  eye. 

The  heat  received  from  the  stars  is  almost  evanescent ; 
only  the  very  slightest  indications  of  it  have  been  rendered 
perceptible  by  the  most  delicate  thermometric  apparatus 
so  far  invented. 

It  is  also  possible  to  obtain  a  very  rough  comparison 
between  the  total  light  actually  emitted  by  the  stars  and  by 
the  sun.  It  is  found,  for  instance,  that  Vega  emits  about 
49  times  the  light  sent  out  by  the  sun.1  Other  stars  give 
similar  results :  we  see,  therefore,  that  our  sun  is  really 
a  rather  faint  star,  but  not  an  infinitesimally  small  one,  in 
comparison  with  Vega. 

There  exists  still  another  very  remarkable  phenomenon 
in  connection  with  stellar  luminosity,  —  its  variability.  For 
it  must  not  be  supposed  that  the  brightness  of  all  stars  is 
strictly  constant :  many  increase  or  diminish  their  light  from 
time  to  time;  and  these  are  called  " variable  stars."  There 
are  several  distinct  kinds.  Some  vary  their  light  steadily, 
ever  increasing  or  diminishing  it ;  others  show  no  regularity 
whatever  in  their  rise  and  fall;  and  a  few,  the  " temporary 
stars,"  or  novce}  appear  suddenly  in  the  heavens,  and  last 
but  a  short  time.  Finally,  there  are  stars  that  wax  and  wane 
with  more  or  less  accurate  periodic  regularity ;  and  wonderful 
variables,  in  which  changes  are  caused  by  some  form  of 
eclipse  phenomenon.  Of  these  last  the  best  known  is  the 
star  called  Algol  (the  Demon). 

1  Note  40,  Appendix. 
326 


STARSHINE 

The  number  of  stars  whose  light  alters  steadily  in  one 
direction  is  very  small :  demonstrated  permanent  variations 
of  brilliancy  since  the  time  of  Hipparchus  are  extremely  in- 
frequent. But  we  have  records  that  Eratosthenes  saw  /3 
Librae  brighter  than  Antares,  though  the  contrary  is  surely 
the  fact  at  present.  Similarly,  in  1603,  Bayer  recorded 
Castor  as  brighter  than  Pollux,  in  the  constellation  Gemini ; 
but  now  Pollux  gives  us  more  light  than  Castor. 

The  most  prominent  irregular  variable  is  77,  in  the  con- 
stellation Argo  Navis,  in  the  southern  celestial  hemisphere. 
Sir  John  Herschel  saw  it  as  bright  as  Sirius  in  1843,  on  the 
occasion  of  a  visit  to  the  Cape  of  Good  Hope.  It  has  been 
of  the  seventh  magnitude  since  1865,  and  is  still  in  process 
of  change. 

Temporary  stars  have  blazed  up  about  eighteen  times 
since  men  began  to  write  their  records  of  the  skies.  The 
most  famous  is  Tycho  Brahe's  star  of  1572,  which  was 
brighter  than  any  other  star,  and  lasted  only  sixteen  months 
all  together.  This  is  the  star  that  first  interested  Tycho  in 
astronomy :  it  is  reported  that  he  refused  for  a  long  time  to 
describe  his  observations,  because  he  thought  it  beneath  the 
dignity  of  a  Danish  nobleman  to  write  a  book. 

A  peculiarly  interesting  recent  object  was  nova  Aurigce 
(the  new  star  in  the  constellation  Auriga) ,  which  appeared  in 
1891.  It  has  been  variously  explained  as  the  result  of 
heat  engendered  by  a  collision  between  two  dark  stars,  or  as 
an  explosion  occurring  in  a  single  dark  body. 

Another  important  new  star  was  found  in  the  constellation 
Perseus  in  1901.  Within  a  few  days  after  discovery  its 
brightness  grew  to  the  first  magnitude,  but  it  faded  again 
with  almost  equal  celerity.  It  developed  a  surrounding 
nebulosity  after  a  time ;  and  is  all  together  one  of  the  most 

327 


ASTRONOMY 

wonderful  astronomic  objects  ever  observed.  The  accom- 
panying Plate  25  shows  the  manner  in  which  this  nebulosity 
grew  in  size.  If  this  growth  is  the  result  of  motion  outward 
from  the  central  star,  the  velocity  must  have  been  incredibly 
rapid.  What  cosmic  process  or  catastrophe  there  occurred 
before  our  eyes,  we  can  neither  describe  fully  nor  attempt  to 
explain. 

The  periodic  variable  stars  are  of  three  kinds.  First, 
those  like  the  star  Mira  (the  Wonder)  in  the  constellation 
Cetus,  which  is  ordinarily  invisible  without  a  telescope, 
but  increases  rather  suddenly  every  eleven  months  to  the 
third  magnitude,  when  it  is  of  course  a  naked-eye  star.  The 
second  variety  of  these,  variables  includes  stars  like  fi,  in  the 
constellation  Lyra,  with  changes  completed  in  a  very  short 
period ;  and  with  the  alteration  of  light  apparently  com- 
pounded of  two  different  variations  superposed.  And  the 
third  class  of  these  stars  are  like  Algol,  in  the  constellation 
Perseus,  which,  at  regular  intervals,  undergoes  a  partial 
eclipse. 

A  possible  explanation  of  regular  variations  might  be  of- 
fered from  the  analogy  of  sunspots.  Certain  stars  may  have 
very  large  permanent  spot  zones  that  are  carried  around  by 
axial  rotation ;  and  when  these  are  turned  toward  the  earth 
we  may  properly  expect  for  a  time  a  considerable  diminution 
of  light. 

The  eclipse  theory  for  the  Algol  stars  is  quite  old ;  but  it 
was  not  proved  correct  in  a  convincing  way  until  1889,  when 
Vogel  successfully  observed  the  spectrum  according  to  the 
Doppler  principle  (p.  284).  He  showed  that  the  visible 
star  Algol  has  a  velocity  of  recession  from  the  earth  before 
its  period  of  minimum  luminosity,  and  an  equal  velocity  of 
approach  after  such  minimum.  The  explanation  now  pos- 

328 


STARSHINE 


tulates  (Fig.  95)  a  dark  but  massive  companion  to  the  visible 
star,  and  both  revolving  in  nearly  circular  orbits  about 
their  common  center  of  gravity.  The  orbit  plane  is  sup- 
posed to  be  directed  towards  the  solar  system,  so  that  we  see 
the  orbits  edgewise.  The  dark  body  is  supposed  to  be 
smaller  in  size  than  the  luminous  or  visible  one. 

In  the  course  of  their  orbital  revolutions,  there  will  come  a 
time  when  the  smaller  dark  body  will  be  at  Z),  and  the  lumi- 
nous body  at  L.     An  observer  on  the  earth,  in  the  direction 
shown  by  the  straight  arrow, 
will  see  the  luminous  body 
partly   covered   or  eclipsed 
by  the  smaller  dark  body. 
As   soon  as   the  eclipse  is 
over,    the    luminous    body 
will  regain  its  full  brilliancy, 
as  in  the  positions  Dr  and  L' '. 

From  a  combination  of 
the  velocities  of  approach 
and  recession,  observed 
spectroscopically,  with  the 
known  light-variations  of 

Algol,  Vogel  was  able  to  prove  that  the  distance  between  the 
two  bodies  is  3f  million  miles,  and  their  diameters  respec- 
tively 0.8  and  1.1  million  miles.  It  will  thus  be  seen  that 
they  are  so  near  each  other  as  to  be  almost  in  rolling  con- 
tact ;  but  there  exists  no  cosmic  law  preventing  the  occur- 
rence of  orbital  motion  of  this  kind. 

The  combined  mass  of  the  two  bodies  Vogel  found  to  be 
about  f  that  of  the  sun;  his  calculations  were  made  by  a 
process  analogous  to  the  method  of  determining  a  planet's 
mass  from  the  observed  orbital  period  of  its  satellite  (p.  204). 

329 


FIG.  95.     Explanation  of  Algol  Stars. 


ASTRONOMY 

The  more  complicated  light-variations  of  certain  other 
stars  may  also  be  explained  on  the  eclipse  theory,  the  orbital 
planes  being  supposed  inclined  to  the  direction  of  the  earth, 
and  neither  body  dark.  If  both  are  luminous,  there  will 
still  be  a  diminution  of  light  during  the  eclipses.  For  one 
luminous  body  will  then  appear  superposed  on  the  other, 
and  the  total  light  will  be  the  same  as  the  larger  body  would 
emit  alone. 

The  next  thing  we  have  to  consider  is  the  important 
question  of  stellar  distances  from  the  solar  system.  After  all, 
this  is  in  a  way  the  most  interesting  matter  we  have  to  dis- 
cuss in  connection  with  the  stars,  for  the  question  at  issue 

really  relates  to  the 
actual  dimensions 
of  the  magnificent 
sidereal  universe. 

In  the  first  place, 
let  us  once  more 
define  "  stellar  par- 

FIG.  96.    Stellar  Parallax. 

Solar  parallax   (p. 

260)  has  been  explained  to  be  half  the  earth's  angular  diam- 
eter, as  seen  from  the  sun.  In  a  similar  way,  stellar  par- 
allax is  the  angle  SS'E,  in  Fig.  96,  between  two  lines,  one 
drawn  from  the  star  S'  to  the  sun  S,  and  the  other  from  the 
star  S',  tangent  to  the  earth's  annual  orbit  around  the  sun 
at  E.  In  other  words,  when  the  earth  reaches  such  a  posi- 
tion in  its  orbit  around  the  sun  that  there  is  a  right  angle  at 
the  earth  E,  between  the  sun  S  and  the  star  S'}  then  the  angle 
SSfE  is  the  star's  parallax. 

We  are  compelled  to  use  the  radius  of  the  earth's  orbit 
in  defining  stellar  parallax,  whereas  we  used  the  radius  of  the 

330 


STARSHINE 

earth  itself  in  the  case  of  solar  parallax.  For  the  earth's  own 
radius  is  far  too  small  to  subtend  an  angle  at  all  appreciable, 
if  the  vertex  is  situated  at  the  vast  distance  of  the  stars. 
Even  when  we  use  the  orbital  radius,  the  very  nearest  star, 
so  far  as  we  now  know,  has  a  parallax  angle  of  only  three- 
quarters  of  a  second  of  arc. 

It  is  clear,  from  Fig.  96,  that  the  star's  parallax  angle 
really  equals  the  difference  in  direction  of  the  star  as  seen 
from  the  earth,  and  as  it  would  be  seen  by  a  supposed  ob- 
server on  the  sun.  As  the  earth  goes  around  its  orbit,  this 
parallactic  displacement,  or  change  of  the  star's  direction 
due  to  the  observer's  being  on  the  earth  instead  of  the  sun  - 
this  parallactic  displacement  must  change  its  direction. 
At  intervals  of  six  months,  during  which  the  earth  traverses 
one-half  of  its  orbit,  the  displacement  reaches  equal  amounts, 
but  opposite  directions.  In  the  interval,  there  is  a  constant 
change  in  the  direction  of  the  displacement ;  so  that,  if  a  star 
is  projected  on  the  sky  at  a  point  perpendicular  to  the  plane 
of  the  earth's  orbit,  it  will  appear  to  describe  in  a  year  a 
little  circle  on  the  sky,  which  is  a  miniature  replica  of  the 
earth's  own  orbit  around  the  sun.1  A  star  in  the  plane  of 
the  earth's  orbit  will  simply  appear  to  swing  back  and  forth 
through  the  year  in  a  short,  straight  line,  instead  of  describing 
a  circle.  Stars  in  intermediate  positions  will  have  apparent 
"parallactic  orbits,"  which  will  be  small  ovals,  intermediate 
in  form  between  the  circle  and  the  straight  line. 
.  The  measurement  of  a  star's  parallax  is  therefore  nothing 
more  than  a  measurement  of  the  form  and  angular  diameter 
of  it's  little  apparent  parallactic  orbit  on  the  sky.  This  may 
be  measured  by  an  " absolute"  method,  or  a  " differential " 

1 A  star  so  situated  is,  of  course,  at  the  "pole"  of  the  ecliptic  circle  on 
the  sky. 

331 


ASTRONOMY 

one.  The  absolute  method  requires  a  determination  of  the 
star's  right-ascension  and  declination  at  various  dates 
during  a  year.  These  being  located  on  a  celestial  chart 
give  the  parallactic  orbit  at  once.  But  the  method  is 
practically  of  little  value,  because  we  possess  no  instruments 
capable  of  measuring  these  declinations,  etc.,  within  the 
small  fraction  of  a  second  of  arc  which  is  here  necessary. 

The  differential  method  is  better,  since  it  enables  us  to  use 
a  micrometer  (p.  276)  as  illustrated  in  Fig.  97.  The  parallactic 
orbit  of  a  star  is  here  shown  as  an  ellipse  or  oval  (p.  331).  Si 
and  82  are  apparent  positions  of  the  star  in  its  parallactic 
orbit  on  two  different  dates.  Si  and  $2'  are 
two  small  stars  in  the  vicinity.  The  observer 
measures,  on  various  dates  through  the  year, 
the  small  angular  distances,  from  the  parallax 
star  to  the  small  stars  Si  and  $2'.  These 
FIG.  97.  Differen-  latter,  on  account  of  their  minuteness,  may 
reasonably  be  supposed  situated  at  practically 
an  infinite  distance  from  us,  and  therefore  to  have  no  appre- 
ciable parallaxes  of  their  own,  and  no  parallactic  orbits.  So 
the  measures  determine  a  series  of  points  on  the  parallactic 
orbit ;  and,  from  the  size  of  the  orbit,  the  parallax  of  the 
star  S. 

It  is  clear  that  this  method  should  bring  out  a  value 
of  the  parallax  possessing  a  high  degree  of  precision : 
and  microscopic  measures  of  an  astronomic  photograph 
may  here  replace  actual  visual  micrometric  work  at 
the  telescope  with  advantage.  If  the  small  stars  are  not 
really  at  an  infinite  distance,  the  differential  method  fur- 
nishes what  may  be  called  " relative  parallaxes";  or,  in 
a  sense,  the  parallax  excess  of  the  star  under  examination 
over  the  small  stars  supposed  infinitely  distant.  The  first 

332 


STARSHINE 

successful  stellar  parallax  measurement  was  made  by  Bessel 
in  1838.  He  used  the  differential  method,  and  applied 
it  to  a  star  of  moderate  magnitude  in  the  constellation 
Cygnus  (cf.  p.  192). 

When  we  come  to  translate  stellar  parallax  measures  into 
terms  of  linear  distances,  We  arrive  at  numbers  so  large  as 
to  be  unmanageable.  For  this  reason  astronomers  have 
invented  a  new  and  large  linear  unit  to  be  used  in  sidereal 
measurements.  It  is  called  the  "light-year,"  and  is  defined 
as  the  linear  distance  light  would  travel  in  a  year.  As  the 
velocity  of  light  is  about  183,000  miles  per  second,  the  light- 
year,  in  miles,  amounts  to  183,000  multiplied  by  the  number 
of  seconds  in  a  year.  It  is  therefore  an  enormous  unit  of 
distance ;  but  it  is  none  too  large  for  use  in  describing  stellar 
distances  in  space.  Its  length  is  about  60,000  times  the  dis- 
tance from  the  earth  to  the  sun,  and  corresponds  to  a  paral- 
lax of  3J  seconds  of  arc. 

Closely  connected  with  the  above  subject  of  stellar  dis- 
tances is  the  question  of  the  stars'  motions.  We  have 
already  seen  (p.  7)  that  the  objects  called  fixed  stars  are 
not  really  fixed  in  space.  They  are  all  actually  drifting 
across  the  sky ;  it  is  only  because  of  their  vast  distance 
that  their  motions  seem  to  us  small  and  slow.  In  reality, 
their  velocities  are  of  the  same  order  of  magnitude  as  those 
existing  among  the  planets  of  our  solar  system :  but  to  the 
rough  instruments  of  the  ancients  these  motions  remained 
unrevealed;  and  so  the  stars  received  their  designation  of 
"fixed,"  to  distinguish  them  from  the  wandering  planets. 
And  within  the  last  half-century  it  has  become  possible 
to  do  even  more  than  merely  measure  this  stellar  drift 
across  the  face  of  the  sky ;  the  drift  which  shows  itself  as  a 
change  of  the  star's  right-ascensions  and  declinations.  As  we 

333 


ASTRONOMY 

have  already  seen  (p.  284),  we  can  now  also  evaluate,  with 
the  spectroscope,  stellar  velocities  of  motion  in  the  "line  of 
sight";  or,  in  other  words,  the  star's  linear  velocities  in  a 
direction  perpendicular  to  the  celestial  sphere. 

Now  changes  in  a  star's  right-ascension  and  declination  do 
not  necessarily  prove  the  existence  of  motion.  For  the  pre- 
cession of  the  equinoxes  (p.  126)  moves  the  point  from  which 
we  count  right-ascensions,  and  it  also  shifts  the  celestial 
equator  from  which  we  count  declinations.  Since  right- 
ascension  is  angular  distance  from  the  vernal  equinox, 
measured  on  the  equator,  and  declination  angular  dis- 
tance from  the  equator,  it  is  clear  that  precessional  changes 
of  equinox  and  equator  will  change  both  these  quantities. 

But  all  precessional  effects  can  be  calculated  easily : 
and  even  after  these  effects  have  been  eliminated  from  our 
observations  by  a  suitable  process  of  calculation,  we  still 
find  small  " residual"  changes  of  right-ascension  and  decli- 
nation. These  may  be  ascribed  wholly  or  in  part  to  real 
motions  of  the  stars.  It  is  this  residual  motion  that  is  called 
a  star's  " proper  motion."  This  term  is  now  a  century  old 
in  astronomy  :  it  is  applied  only  to  motion  across  the  celestial 
sphere;  not  to  motion  in  the  line  of  sight,  revealed  spec- 
troscopically.  The  latter  has  been  separately  denominated 
" radial  velocity."  Proper  motion  is  measured  in  seconds  of 
arc  per  annum ;  radial  velocity  in  linear  miles  per  second. 

Less  than  two  hundred  stars  have  proper  motions  as  large 
as  one  second  of  arc  per  annum.  The  largest  known  mo- 
tion of  the  kind  belongs  to  a  little  star  numbered  V  243  in  a 
great  catalogue  of  stars  made  at  Cordova  in  South  America. 
It  drifts  nearly  nine  seconds  annually.  The  next  largest  and, 
until  1898,  the  largest  known  belongs  to  a  star  numbered 
1830  in  a  catalogue  observed  by  Groombridge  in  England 

334 


STARSHINE 


about  the  year  1790.    This  star  is  often  called  the  "  run- 
away" ;  it  travels  seven  seconds  in  a  year. 

The  relation  of  proper  motion  and  radial  velocity  is  indi- 
cated in  Fig.  98.  If  a  star  at  Si  moves  to  $2  in  a  unit  of  time 
(a  year,  let  us  say),  and  if  the  solar  system  is  at  E,  we  can 
draw  SiSi  perpendicular  to  SiE,  to  indicate  the  star's  appar- 
ent motion  across  the  celestial  sphere  as  seen  from  E.  In  the 
same  unit  of  time  the  star  will  have  receded  from  the  earth  by 
a  distance  SiS2'',  and  this  will 
represent  its  annual  radial  ve- 
locity. The  true  motion  of  the 
star,  SiS2,  may  be  regarded  as 
really  made  up  of  two  parts, 
-the  radial  motion  and  the 
proper  motion.  Now  that  we 
are  able  to  measure  these 
radial  velocities,  it  is  at  last 
possible  to  ascertain  from  ob- 
servation both  parts  of  the  dis- 
tance SiSz,  which  is  the  star's 
true  motion  in  a  unit  of  time. 
Before  we  had  the  spectro- 
scope and  Doppler's  principle,  we  never  knew,  or  could  know 
more  than  the  one  part  SiSi.  These  considerations  empha- 
size the  importance  of  the  spectroscope  in  sidereal  research : 
it  has  not  only  created  a  celestial  chemistry ;  it  has  also  given 
us  new  and  essential  knowledge  in  the  oldest  department  of 
astronomic  science,  —  the  theory  of  sidereal  motions. 

Not  infrequently  it  happens  that  a  certain  number  of 
stars  are  found  to  have  proper  motions,  and  probably  real 
motions  in  space,  nearly  identical,  both  in  amount  and  direc- 
tion. For  instance,  Proctor  pointed  out  some  years  ago 

335 


FIG.  98. 


Proper  Motion  and  Radial 
Velocity. 


ASTRONOMY 

that  five  of  the  bright  stars  in  the  Dipper  (p.  52)  probably 
possess  such  a  community  of  motion.  There  can  be  little 
doubt  that  they  are  proceeding  through  space  in  parallel 
lines,  and  that  they  belong  together.  The  same  is  true  of  a 
number  of  the  brighter  stars  in  the  Pleiades  group.  They 
are  not  merely  an  apparent  group,  but  a  real  cluster. 

A  very  surprising  thing,  discovered  in  1909,  is  that  Sirius 
is  also  probably  a  member  of  the  Dipper  group  of  stars.  If 
this  be  so,  the  group  in  question  is  not  a  distant  congeries 
passing  us  at  a  distance,  perhaps,  of  billions  of  miles;  but 
it  is  passing  so  close  that  we  are  actually  now  within  the 
drifting  group  of  stars  itself.  This  follows  from  the  fact 
that  Sirius  is  in  quite  a  different  part  of  the  sky  from  the 
Dipper  region. 

We  have  seen  quite  enough  to  make  clear  the  high  value 
to  science  of  these  very  modern  measures  of  radial  velocity : 
unfortunately,  it  has  not  been  found  possible  as  yet  to  apply 
the  method  to  faint  stars,  because  their  light  is  not  sufficient 
to  give  a  measurable  spectral  image.  This  class  of  work  was 
first  attempted  by  Huggins  in  1867  ;  and  he  began  by  measur- 
ing the  radial  velocity  of  Sirius,  the  brightest  of  the  stars, 
with  a  visual  telescope  and  spectroscope.  It  was  not  until 
Vogel  began  to  use  photography  in  1888  that  any  consider- 
able extension  of  the  process  became  possible.  Fainter 
stars  then  first  became  observable,  for  the  exposure  of  a 
photographic  plate  can  be  lengthened  within  reasonable 
limits,  so  as  to  give  even  a  small  quantity  of  light  time  to 
impress  a  spectral  image  on  the  plate. 

Good  examples  of  radial  velocities  are  found  in  the  star 
a  Carinse,  receding  from  the  solar  system  17  miles  per 
second ;  and  o  Ceti,  receding  54  miles  per  second.  As  the 
earth's  own  orbital  velocity  around  the  sun  is  about  18| 

336 


STARSHINE 

miles  per  second,  we  see  that  the  stellar  motions  are  only  of 
the  same  order  of  velocity  as  those  existing  within  the  con- 
fines of  the  solar  system. 

Spectrum  photographs  of  the  above  stars  are  shown  in  the 
lower  part  of  Plate  26.  In  each  case  the  stellar  spectrum 
is  placed  between  two  artificial  spectra,  produced  in  the 
observatory  for  comparison.  It  will  be  seen  that  the  stellar 
spectral  lines  of  both  stars  are  displaced  in  the  same  direction 
with  respect  to  the  artificial  spectra,  because  both  stars 
are  receding;  but  the  displacement  is  much  greater  in  the 
case  of  o  Ceti,  on  account  of  its  greater  velocity  of  recession. 
The  upper  photograph  of  Plate  26  is  a  comparison  of  the  lunar 
spectrum  with  that  of  Saturn  and  its  ring  (cf.  p.  245).  It 
shows,  as  it  should,  that  the  outer  part  of  the  ring  is  moving 
slower  than  the  inner  part. 

This  matter  of  stellar  spectroscopy  was  first  taken  up  by 
Huggins  to  study  the  chemical  composition  of  the  stars, 
though  it  led  him  also  to  the  measurement  of  radial  veloci- 
ties, as  we  have  seen.  At  about  the  same  time,  Secchi 
undertook  similar  investigations,  and  to  him  we  owe  a  sort 
of  classification  of  stellar  spectra,  as  follows : 

1.  White  and  blue  stars,  with  strong  evidence  of  hydrogen. 
Examples  are  Sirius  and  Vega.     These  stars  are  believed  to 
be  in  an  early  stage  of  cosmic  development. 

2.  Solar  stars,  showing  in  their  spectra  many  dark  lines 
due  to  absorption,   as  in  the  solar  spectrum.     Capella  is 
one  of   those  stars ;    they  are  supposed   to  be   somewhat 
older  than  those  showing  the  hydrogen  lines. 

3.  Red  stars,  with  spectra  showing  dark,  broadened  lines. 

4.  Faint  red  stars,  probably  very  old ;  the  spectra  having 
a  few  bright  lines. 

All    the    spectroscopic    observations    indicate    a    stellar 
z  337 


ASTRONOMY 

chemistry  similar  to  that  of  the  solar  system.  The  entire 
sidereal  universe  seems  to  contain  but  one  set  of  chemical 
elements;  and  these  are  very  widely  distributed.  So  we 
see  once  more  that  our  sun  is  a  star;  and  if  the  other 
stars  are  traveling  through  space,  our  sun  should  also  be  in 
motion,  carrying  its  planets  with  it,  much  as  the  earth 
moves  in  its  orbit  around  the  sun  and  carries  the  moon  with 
it.  Analogy  would  lead  us  to  expect  such  a  solar  motion. 

Of  course,  the  simplest  way  to  study  this  question  is  to 
examine  the  radial  velocities  of  a  large  number  of  stars.  Sup- 
pose we  find  that  those  near  a  certain  point  on  the  sky  are 
approaching  us;  those  near  the  opposite  point  receding; 
and  those  halfway  between,  neither  approaching  nor  reced- 
ing. Then  we  may  conclude  that  this  is  merely  a  result  of 
the  solar  system's  own  motion ;  and  that  we  are  approaching 
the  stars  projected  near  the  point  of  the  sky  where  they 
seem  to  be  approaching  us.  Towards  this  point  on  the  sky, 
then,  the  solar  motion  is  for  the  moment  directed.1 

Campbell  made  such  a  research  a  few  years  ago,  using 
spectroscopic  results  derived  from  280  stars.  The  point 
on  the  sky  indicated  by  his  work  is  not  very  far  from  the 
first-magnitude  star  Vega.  It  is  called  the  "apex  of  the 
sun's  way."  Naturally,  Campbell  used  only  bright  stars 
whose  spectra  could  be  observed ;  and  of  course  brightness 
indicates  nearness,  other  things  being  equal.  It  may  there- 
fore be  a  fact  that  stars  having  a  common  drift  with  the  sun 
predominate  in  Campbell's  series  of  observations ;  and  if  so, 
this  might  partly  invalidate  his  result.  But  however  this 
may  be,  he  finds  the  region  near  Vega  to  contain  the  apex, 
and  13  miles  per  second  as  the  " cosmic  linear  velocity"  of 
the  solar  system. 

1  Note  41,  Appendix. 
338 


STARSHINE 

Another  fact  that  may  cause  a  slight  error  in  such  an  in- 
vestigation as  the  foregoing  is  the  necessity  of  making 
some  assumption  as  to  the  average  real  motions  of  the  stars 
whose  radial  velocities  are  observed.  For  we  do  not  find  all 
stars  near  the  apex  approaching  us ;  only  a  preponderance 
of  motions  of  approach.  So  astronomers  assume  that  in 
the  average  of  so  large  a  number  of  stars  (in  this  case,  280) 
there  will  be  as  many  motions  in  any  one  direction  as  in  any 
other.  Therefore,  a  preponderance  of  motions  of  approach 
near  the  apex  must  be  due  to  solar  motion,  not  to  motion  of 
the  stars  themselves. 

It  is  singular  that  as  far  back  as  1783  Sir  William  Herschel 
obtained  almost  the  same  result  from  a  discussion  of  the 
proper  motions  (p.  334)  of  various  stars  across  the  sky.  His 
method  is  well  illus- 
trated in  Fig.  99.  If 
there  are  two  lamp- 
posts, LI  and  Z/2,  on 
opposite  sides  of  a 

. ,  ,         -,.         FIG.  99.    Determination  of  the  Apex  (Herschel). 

street,  the  angular  dis- 
tance between  them  will  seem  much  larger  when  they  are 
viewed  from  HI  than  from  Hi.  A  person  walking  from  HI 
to  H2  will  see  this  increase  of  the  angular  distance.  Apply- 
ing this  principle  to  the  sky,  Herschel  concluded  that  near 
the  apex  the  constellations  must  be  opening  out,  as  we  ap- 
proach, and  at  the  opposite  point  of  the  sky  they  must 
be  closing  in. 

In  other  words,  near  the  apex  stellar  proper  motions  directed 
away  from  that  point  must  predominate ;  and  near  the  op- 
posite point,  called  the  " anti-apex,'7  proper  motions  directed 
towards  the  critical  spot  must  be  most  in  evidence.  Herschel 
had  at  his  disposal  the  measured  proper  motions  of  only 

339 


ASTRONOMY 

thirteen  stars  all  together.  Yet  with  the  insight  of  rare 
genius,  he  so  sifted  this  meager  evidence  that  he  was  able 
to  find  the  right-ascension  and  declination  of  the  apex 
with  some  approximation  to  correctness. 

Later  investigators  have  of  course  repeated  this  work  with 
much  more  elaborate  modern  material  at  command.  They 
find  a  result  in  very  fair  accord  with  the  spectroscopic  one. 
But  they  also  find  this  important  peculiarity.  If  the  proper 
motions  are  divided  into  groups,  and  the  calculations  made 
separately  with  stars  of  large  and  small  proper  motions, 
the  apex  comes  out  farther  south  on  the  sky  for  the  stars  of 
large  proper  motions. 

Now  it  is  evident  that  any  investigation  of  this  kind  must 
assume  that  if  there  were  no  solar  motion,  the  stellar  proper 
motions  would  be  quite  casual,  and  free  from  any  tendency 
to  congregate  in  direction  towards  or  from  any  apical  point. 
But  such  a  tendency  in  the  stars  themselves  is  indicated 
by  the  peculiar  result  just  mentioned.  It  is  clear  that  the 
large  proper  motion  stars  must  have  a  common  path  of 
their  own.  But  largeness  of  proper  motion  should  indicate 
nearness  to  us,  other  things  being  equal ;  for  at  a  sufficient 
distance,  even  large  proper  motions  would  shrink  into  ap- 
parent nothingness.  Therefore  it  is  within  the  bounds  of 
possibility  that  our  sun  is  a  member  of  a  drifting  stream  of 
stars,  to  which,  in  general,  the  large  proper  motion  stars 
belong  also. 

In  the  light  of  the  above  discussion  of  cosmic  motions  of  the 
sun  and  stars,  as  well  as  stellar  distances,  it  is  possible  to  con- 
sider an  interesting  special  problem  which  may  be  solved 
approximately  with  modern  observational  data.  We  have 
seen  that  the  solar  system  is  moving  toward  Vega  at  the 
rate  of  13  miles  per  second  (p.  338).  Observations  of  Vega's 

340 


STARSHINE 

own  radial  velocity  indicate  that  it  is  itself  receding  from  us 
at  the  rate  of  3  miles  per  second ;  so  we  are  overtaking  it  at 
the  rate  of  10  miles  per  second.  But  the  number  of  seconds 
in  a  year  is,  approximately,  365  X  24  X  60  X  60 ;  and  the 
actual  annual  approach  of  the  two  stars,  therefore,  365  X  24 
X  60  X  60  X  10  miles.  The  parallax  of  Vega  has  been 
measured;  it  is  O."ll.  From  the  parallax,  the  distance 
between  the  solar  system  and  Vega  may  be  computed,1  and 
it  comes  out,  approximately : 

93000000  X  200000 

^rr 

To  ascertain  the  time  in  years  required  by  the  solar 
system  to  reach  the  position  occupied  by  Vega,  we  must 
divide  the  distance  of  Vega  by  the  rate  of  approach  per  year. 
We  thus  obtain,  for  the  number  of  years  required  to  over- 
take Vega  in  space : 

93000000  X  200000  _1_  _, 

0.11  <  365  X  24  X  60  X  60  X  10 

or,  approximately,  560,000  years ;  and  after  the  lapse  of  that 
period  of  time,  the  solar  system  should  reach  Vega. 

But  in  that  interval  Vega  will  have  moved,  too,  for  it 
has  a  proper  motion  across  the  sky  of  0."5  per  year,  which  is 
about  four  times  its  parallax  angle.  Figure  100  will  explain 
this  state  of  affairs.  At  a  certain  moment,  the  sun  and 
earth  are  at  S  and  E,  with  Vega  at  Vi.  Then,  by  definition, 
the  angle  EViS  is  Vega's  parallax.  At  the  end  of  a  year, 
the  earth  will  be  back  at  E,  very  nearly,  but  Vega  will  be 
seen  at  Vz,  because  its  proper  motion  will  have  carried  it 
across  the  sky,  as  seen  from  the  solar  system,  through  the 
angle  Fi$F2.  Since  this  angle  is  four  times  the  parallax 

1  Note  42,  Appendix. 
341 


ASTRONOMY 

angle  EViS,  it  must  follow,  approximately,  that  FiF2  is 
four  times  ES.  Therefore,  ViV2  is  4  X  93,000,000  miles,  or 
372,000,000  miles. 

So  we  see  that  when  the  sun  reaches  the  point  where 
Vega  should  be  in  560,000  years,  Vega  will  have  moved  at 


FIG.  100.     Vega's  Parallax  and  Proper  Motion. 

least  372,000,000  X  560,000  miles,  and  this  is  about  as 
near  as  we  shall  ever  approach  Vega.  What  will  be  Vega's 
parallax  at  that  time?  We  can  answer  this  question  by 
comparing  the  present  distance  of  Vega,  which  we  have 
found  to  be : 

93000000  X  200000 

-air       miles' 

with  its  distance  in  560,000  years  as  just  obtained.  The 
two  numbers  are  not  far  from  equal,  and  therefore  Vega's 
future  parallax  will  not  be  far  from  its  present  parallax. 
So  there  is  no  danger  of  a  cosmic  collision  with  Vega,  so  far 
as  we  may  judge  from  the  above  rough  calculation. 

Our  present  discussion  would  not  be  complete  without  a 
brief  account  of  certain  quite  recent  researches,  made 
principally  by  Kapteyn.  His  idea  is  that  we  need  extensive 
statistical  knowledge  of  stellar  distribution,  more  than  direct 
measures  of  a  few  parallaxes.  He  therefore  undertakes  to 
compute  the  average  parallax  of  the  stars  of  any  given  magni- 

342 


STARSHINE 

tude,  separating  the  stars  thus  by  magnitudes  for  the  obvious 
reason  that  the  fainter  ones  may  be  expected  to  average 
greater  distances  than  the  brighter  ones,  and  therefore 
smaller  parallaxes.  Figure  101  shows  his  method  of  attack 
upon  this  problem.  The  arrow  SS'  is  intended  to  represent 
the  annual  proper  motion  of  a  star  S  across  the  face  of  the 
sky,  the  arrow,  of  course,  indicating  both  the  direction  and 
quantity  of  such  motion.  The  position  of  the  apex  on  the 
sky  is  shown  at  A.  The  line 
S'Si  is  drawn  perpendicular 
to  SA,  and  the  smaller  arrow 
SSi  then  shows  how  much 
the  proper  motion  SS'  carried 
the  star  away  from  the  apex. 
In  fact,  the  proper  motion 
SS'  may  be  regarded  as  com- 
pounded of  two  motions : 

.  FIG.  101.    Kapteyn's  Researches. 

&oi ,  which  affects  the   an- 
gular distance  from  the  star  to  the  apex;  and  Si'S',  which 
does  not  affect  that  distance,  being  at  right  angles  to  SA. 

Now  Kapteyn  " resolves"  (as  it  is  called)  all  known 
proper  motions  into  two  such  "  components,"  one  directed 
away  from  the  apex,  the  other  at  right  angles  to  the  first. 
But  we  have  already  seen  (p.  339)  that  the  effect  of  the  solar 
system's  own  motion  in  space  is  to  open  out  the  constella- 
tions near  the  apex ;  therefore  SSi,  the  star's  proper  motion 
component  away  from  the  apex,  must  include  the  effect 
of  the  solar  system's  motion ;  but  Si'S',  the  other  component, 
is  free  from  such  effect.  In  the  general  average  of  a  large 
number  of  stars  of  any  given  magnitude,  the  two  components 
should  be  equal,  if  the  sun  were  at  rest.  For  the  effect  of 
the  solar  motion  would  then  disappear ;  and  we  may  assume 

343 


ASTRONOMY 

that  the  star's  own  motions  are  as  likely  to  be  in  one  direc- 
tion as  another,  so  that  the  average  components  would 
balance,  as  it  were.  It  follows  that  any  difference  of  the 
two  components,  derived  from  observed  averages,  must  be 
an  effect  of  the  sun's  motion  alone. 

Now  this  average  difference  is  expressed  in  seconds  of  arc, 
being  an  observed  angular  proper  motion  across  the  sky. 
Arid  we  know  the  linear  velocity  of  the  solar  system  toward 
the  apex  to  be  13  miles  per  second  (p.  338),  or  13  X  365  X  24 
X  60  X  60  miles  annually.  Figure  102  shows  the  further 


FIG.  102.    Kapteyn's  Researches. 

procedure.  The  arrow  SiS2  represents  the  solar  system's 
motion  toward  the  apex,  in  a  year.  Kapteyn's  "  average 
star"  is  shown  at  Ss.  The  little  angle  SiSA  is  the  average 
star's  proper  motion  component  away  from  the  apex,  or 
the  above-mentioned  observed  difference  of  the  two  com- 
ponents. Knowing,  now,  this  little  angle,  and  the  linear 
velocity  SiSz,  we  can  calculate  SiSz,  or  the  average  star's 
distance  from  the  solar  system  Si. 

This  beautiful  method  of  investigation  has  enabled 
Kapteyn  to  obtain  a  table  of  approximate  stellar  distances. 
He  gives  the  following  results,  for  two  types  of  stars  sepa- 
rately (cf .  p.  337) :  I,  bluish  white  stars,  like  Sirius ;  II,  solar 
stars,  like  Capella.  The  distances  are  expressed  in  light- 
years  (p.  333),  three  of  which  correspond  approximately  to 
a  parallax  of  one  second  of  arc.  The  table  shows  that  the 

344 


STARSHINE 


solar  stars  are  decidedly  nearer  to  us  than  are  the  Sirian 
stars. 

TABLE  OF  KAPTEYN'S  DISTANCES 

(In  light-years) 


STAB'S  MAGNITUDE 

TYPE 

I 

II 

1 

101 

43 

2 

130 

56 

3 

166 

71 

4 

213 

91 

5 

273 

117 

10 

948 

405 

15 

3270 

1404 

Kapteyn  has  also  obtained  some  further  interesting  results 
as  to  stellar  distribution.  He  uses  a  "unit  of  sidereal  space/' 
and  for  this  space  unit  he  imagines  a  cosmic  sphere  of  "unit 
radius,"  which  he  defines  as  a  sphere  such  that  a  star  on 
its  surface  would  have  a  parallax  of  one  second  of  arc  to 
an  observer  at  its  center.  The  length  of  this  radius  would 
be  about  three  light-years.  Then,  since  parallax  angles  are 
inversely  proportional  to  distances,  it  follows  that  stars 
whose  parallaxes  are  greater  than  O."20  are  all  within  a 
sphere  whose  radius  is  5  units.  About  20  stars  with  such 
parallaxes  are  known;  and  we  may  assume  that  these  are 
probably  all  that  exist. 

Now  the  volumes  of  spheres  are  proportional  to  the 
cubes  of  their  radii ;  so  that  the  above-mentioned  sphere  with 
radius  5  must  have  a  volume  of  125.  And  as  there  are  20 
stars  in  it,  there  must  exist  about  one  star  to  each  six  units 
of  space;  and  this  is  the  approximate  "star-density"  in  the 
cosmic  vicinity  of  the  solar  system.  This  conclusion  may 

345 


ASTRONOMY 

not  be  very  accurate;  but  it  constitutes  a  most  important 
addition  to  our  knowledge  of  sidereal  astronomy,  since  even 
a  rough  approximation  is  better  than  a  total  absence  of 
information. 

And  Kapteyn  has  been  able  to  proceed  a  step  farther. 
Having  found  the  average  distance  of  stars  of  any  given 
magnitude,  and  knowing  the  average  proper  motions  and 
radial  velocities  as  well,  he  has  been  able  to  compute  the 
actual  average  linear  velocities  of  the  stars  in  space;  and 
he  finds  them  to  be  somewhat  less  than  twice  the  cosmic 
velocity  of  the  solar  system.1  Therefore  the  stars'  average 
speed  is  about  26  miles  per  second;  or,  in  a  year,  about 
seven  times  the  distance  between  the  earth  and  sun.  But 
the  sidereal  unit,  or  radius  of  the  unit  sphere,  is  about 
200,000  times  the  distance  from  earth  to  sun ;  so  the  stars 
move  a  sidereal  unit,  on  the  average,  in  27,000  years. 

Now  we  have  found  that,  on  the  average,  about  one  star 
exists  in  each  six  units  of  space.  From  this  it  may  be  com- 
puted, according  to  the  theory  of  probabilities,  that  the 
average  distance  between  the  stars  is  about  3.5  linear 
sidereal  units.  Therefore  the  stars  move  through  a  distance 
equal  to  the  3,5  units  that  separate  them  in  about  3.5  X 
27,000,  or  100,000  years,  on  the  average.  It  follows  from 
all  these  considerations  that  stars  will  approach  each  other 
infrequently,  even  within  astronomical  proximities,  enormous 
as  these  are. 

The  above  conclusions  all  relate  to  averages ;  and  we  know 

1  Knowing  the  star's  parallax,  or  distance,  and  the  angular  annual 
proper  motion  across  the  sky,  we  compute  the  linear  velocity  component 
parallel  to  the  celestial  sphere  in  the  same  way  that  we  obtain  a  planet's 
linear  diameter  from  its  measured  angular  diameter.  Then,  knowing  the 
radial  velocity,  and  the  linear  velocity  at  right  angles  to  it,  as  computed 
from  the  proper  motion,  we  can  finally  calculate  the  actual  velocity. 

346 


STARSHINE 

one  or  two  stars  that  probably  have  far  greater  velocities 
than  the  average.  For  instance,  the  star  1830  Groombridge 
(p.  334)  has  a  velocity  of  perhaps  140  miles  per  second.  We 
must  conclude  that  this  particular  star  is  passing  through 
our  sidereal  universe,  and  will  leave  it  altogether  in  a  few 
million  years.  But  in  general,  from  what  has  been  said, 
it  would  appear  that  the  stars  in  our  universe  are  much  like 
the  molecules  of  a  gas,  as  indicated  in  the  kinetic  theory  of 
gases.  The  difference  between  a  glass  vessel  full  of  gas  and  a 
universe  full  of  stars  is  merely  one  of  scale.  In  either  case, 
each  star  or  molecule  moves  in  a  more  or  less  straight  line 
of  random  direction,  until  or  unless  a  couple  of  them  happen 
to  collide.  In  both  cases,  such  collisions  are  extremely  fre- 
quent :  only,  in  the  gas,  the  word  " frequent"  signifies  a  very 
minute  fraction  of  a  second ;  in  stellar  space,  the  same  word 
may  mean  centuries. 

Perhaps  this  kinetic  theory  of  stars  would  undergo  some 
material  modification  if  we  admit  that  all  observed  stellar 
motions  are  not  necessarily  random  ones,  and  that  there  may 
be  star-groups  of  common  motion,  and  star-streams  of  vast 
extent.  But  however  this  may  be,  these  magnificent 
researches  are  all  inspiring  in  a  high  degree :  it  is  extraordi- 
nary that  such  can  still  be  made  in  our  own  day  in  the  oldest 
and  most  completely  perfected  domain  of  human  knowledge, 
the  science  of  astronomy. 

Not  infrequent  in  the  sky  are  the  " binary"  stars;1  twin 
suns,  they  have  been  called  (cf.  p.  9).  Each  component  star 
of  such  a  pair  moves  around  the  center  of  gravity  of  both  in  an 
oval  orbit,  just  as  the  earth  and  moon  (p.  174)  move  around 
their  center  of  gravity.  The  orbits  are  studied  with  a  special 
micrometer  (p.  276),  with  which  astronomers  can  measure 

1  The  lower  part  of  Plate  27  is  a  photograph  of  a  binary. 
347 


ASTRONOMY 

the  angular  distance  between  the  two  components  in  seconds 
of  arc,  and  also  the  angle  between  the  line  joining  them  on 
the  sky  and  a  line  drawn  from  the  principal  one  to  the  celes- 
tial pole.  Thus,  in  Fig.  103,  SiP  is  an  arc  of  a  great  circle 
imagined  on  the  sky,  joining  the  principal 
component  star  Si  and  the  pole.  We  then 
measure  the  small  angular  distance  SiSzt  be- 


tween the  two  components  of  the  double  star, 
'  and  also  the  angle  PSiSz.     The  latter  is  called 

the  " position  angle."     If  we  continue  these 
FIG.  IDS.   Binary    measures,  at  intervals,  for  a  number  of  years, 

and  then  draw  the  resulting  orbital  curve,  we 
find  it  to  be  an  oval  or  ellipse,  often  very  much  flattened. 
But  this  is  only  an  apparent  orbit ;  for  what  we  see  is  the 
real  orbit,  projected  on  the  celestial  sphere.  It  is  only  when 
the  plane  of  the  binary  star's  orbit  happens  to  be  perpen- 
dicular to  our  sight-line  that  the  apparent  orbit  coincides  with 
the  real  one.  But  it  is  always  possible  to  calculate  the 
location  of  the  true  orbital  plane ;  and,  in  fact,  the  elements 
of  the  real  orbit,  by  applying  Kepler's  laws  of  motion  (p.  187) 
to  the  apparent  orbit.  Yet,  even  after  this  has  been  done,  we 
have  only  a  "relative"  orbit,  representing  the  motion  of 
one  component  star  of  the  binary  pair  with  respect  to  the 
other. 

For  a  very  few  binaries,  the  actual  orbit  of  both  compo- 
nents has  been  separately  determined,  by  means  of  micro- 
metric  comparisons  with  neighboring  small  stars.  And 
when  we  are  so  fortunate  as  to  know  also  the  parallax  of 
the  binary,  we  can  calculate  the  linear  dimensions  of  the 
orbits  in  miles.  Without  the  parallax,  we  can  know  only 
the  angular  dimensions  of  these  orbits  in  seconds  of  arc. 
The  relation  of  the  two  dimensions  is  like  the  relation  of  the 

348 


Photo  by  Barnard.  Photo  Mt.  Wilson  Observatory. 

PLATE  27.     A  Star  Cluster  in  Hercules  and  the  Double  Star  Krueger  60. 


STARSHINE 

angular  diameter  of  the  sun  to  its  real  linear  diameter 
(p.  118).  In  the  few  cases  for  which  such  researches  have 
been  carried  out  with  success,  the  linear  size  of  the  orbits 
appears  to  be  comparable  with  the  orbital  radii  we  find  in 
the  solar  system.  So  we  conclude  that  the  binary  systems 
are  not  extremely  large,  speaking  cosmically. 

Certain  binary  stars  have  been  recognized  as  such  by 
spectroscopic  instead  of  micrometric  observations.  We  have 
already  described  Vogel's  discoveries  with  regard  to  Algol 
(p.  328),  where  the  binary  character  of  the  star  was  betrayed 
by  an  observed  periodic  change  in  the  direction  of  its  radial 
velocity.  But  Pickering  also  found  that  certain  spectra 
photographed  with  a  slitless  spectroscope  (p.  285)  showed 
a  periodical  doubling  of  the  lines.  Ordinarily  single,  they 
became  double  at  uniform  intervals  of  time.  Pickering 
explained  this  correctly  as  indicating  a  binary  system,  in 
which,  unlike  Algol,  the  components  are  both  luminous  stars. 
When  one  component  is  approaching  us,  and  the  other  reced- 
ing, in  consequence  of  their  orbital  motions,  the  spectral  lines 
are  displaced  in  opposite  directions,  according  to  the  Doppler 
principle,  and  we  get  two  separate  spectra  and  two  sets  of 
lines.  When  both  components  are  moving  at  right  angles 
to  the  sight-line,  as  they  will  do  in  another  part  of  their 
orbit,  the  two  spectra  are  superposed,  and  we  get  one  set  of 
lines  only.  It  is  remarkable  that  we  can  thus  separate  a 
pair  of  stars  with  certainty,  although  they  appear  so  near 
each  other  on  the  sky  that  the  most  powerful  telescope  shows 
but  a  single  object  at  all  times.  This  was  a  great  triumph 
for  the  spectroscope ;  the  observation,  together  with  the 
correct  explanation  of  it,  will  surely  have  a  place  in  the 
classic  annals  of  astronomy. 

It  is  interesting  to  note  that  we  can  calculate  also  the 

349 


ASTRONOMY 

masses  of  these  binary  stars  for  which  both  orbits  and  paral- 
laxes have  been  observed.  As  we  have  just  seen,  we  then 
know  the  linear  dimensions  of  the  orbit  in  miles,  and  so  we 
can  apply  the  method  used  (p.  204)  for  obtaining  the  mass  of 
a  planet  having  an  observable  satellite.1  The  masses  of 
these  binaries,  in  the  few  cases  where  they  have  become 
known,  are  found  to  be  of  the  same  order  of  magnitude  as  the 
sun's  mass. 

Before  leaving  the  subject  of  binary  stars,  it  may  be  of 
interest  to  touch  on  one  possible  theory  as  to  their  origin. 
It  is  not  now  believed  by  astronomers  that  the  Laplacian 
theory  of  celestial  development  (p.  235)  is  the  only  pos- 
sible or  even  probable  one.  For  the  Laplacian  idea  leads 
to  a  single  central  sun,  with  many  planets  of  far  smaller 
size  than  the  sun.  But  it  is  possible  that  an  original 
whirling  nebula  may  have  undergone  changes  more  or  less 
approximating  the  formation  of  two  nuclei.  These, 
revolving,  gave  rise,  first,  to  an  egg-shaped,  —  later,  a 
dumb-bell  shaped,  —  revolving  body.  The  latter,  finally 
separating,  should  produce  twin  suns,  at  first  revolving  with 
their  surfaces  almost  in  contact.  Such  a  condition  might 
even  explain  some  of  the  peculiar  light-changes  of  certain 
variable  stars.  Later,  there  might  arise  perturbative 
action,  similar  to  the  tidal  effects  produced  between  the  moon 
and  terrestrial  oceans.  These  would  drive  the  two  bodies 
farther  apart  (cf .  p.  258),  and  possibly  lead  to  a  visible  binary 
star.  Nor  is  there  any  objection  to  our  imagining  some  of 
these  distant  suns  to  be  attended  by  planets.  Only,  if 
such  planets  are  no  larger  than  Jupiter,  we  could  not  possibly 
hope  to  see  them  with  the  most  powerful  of  our  telescopes. 

It  is  an  easy  transition  from  the  binary  systems  to  those 

1  Note  43,  Appendix. 
350 


PLATE  28.     The  Pleiades. 


Photo  by  Barnard. 


STARSHINE 

still  more  wonderful,  in  which  three  or  more  incandescent 
suns  revolve  about  their  common  center  of  gravity  in  plain 
view  of  the  telescope.  For  instance,  the  constellation  Lyra 
contains  an  important  double  star,  the  "  double  double/ ' 
in  which  each  component  is  itself  a  binary,  forming  all  to- 
gether a  quadruple  star. 

And  in  addition  to  these  " multiple  stars,"  we  find  also 
various  "  clusters."  Some  contain  comparatively  few  stars, 
spread  over  quite  a  considerable  bit  of  the  sky.  The 
Pleiades  group  is  a  famous  cluster  of  this  kind.  Two  views 
of  it  are  shown  in  the  accompanying  Plate  28 :  one  con- 
tains the  stars  only ;  the  other,  made  with  a  large  telescope, 
indicates  that  most  of  these  stars  are  still  surrounded  with 
nebulosities.  Here  and  there  we  can  see  a  nebulous  lane 
running  from  one  star  to  the  next ;  nor  have  these  peculiar 
formations  ever  been  thoroughly  explained.  There  are 
also  other  clusters,  like  the  close-packed  globular  one  in 
the  constellation  Hercules  (Plate  27,  p.  349),  consisting  of 
many  thousand  stars  separated  from  each  other  on  the  sky 
by  very  small  angular  distances  only. 

We  have  two  sure  facts  to  indicate  that  the  clusters  are 
single  objects,  and  not  mere  fortuitous  groupings  of  stars, 
unconnected  with  each  other,  situated  at  all  sorts  of  distances 
from  the  solar  system,  and  appearing  close  together  because 
they  happen  to  be  projected  near  a  single  point  of  the  sky. 
First,  in  the  Pleiades  group,  it  is  known  that  over  forty  stars 
have  practically  identical  proper  motions  in  the  same  direc- 
tion on  the  sky,  pointing  to  a  community  of  real  motion  in 
space.  And  secondly,  many  close  clusters  have  been  found 
to  contain  a  most  unusually  large  percentage  of  variable 
stars,  again  indicating  a  community  of  origin  for  the  whole 
cluster. 

351 


ASTRONOMY 

Unfortunately,  it  has  not  yet  been  possible  to  measure 
the  parallax  of  any  cluster.  We  can  but  make  guesses  at 
their  distance  from  the  solar  system,  rough  estimates  based 
on  the  size  of  their  proper  motions.  These  are  so  small  that 
we  must  assume  the  clusters  to  be  very  distant, — probably  not 
less  than  400  light-years.  If  removed  to  that  distance,  our 
own  sun  would  give  us  no  more  light  than  a  star  of  the 
eleventh  magnitude.  It  follows  that  the  clustered  stars 
are  perhaps  comparable  in  size  with  the  sun,  for  they,  too, 
average  the  eleventh  magnitude,  more  or  less. 

With  the  above  estimate  of  distance,  we  can  also  estimate 
the  linear  size  of  the  clusters  from  their  angular  diameter, 
in  the  usual  way ;  and  we  find  them  to  be  about  two  light- 
years  in  diameter.  If  such  a  cluster  contained  10,000  stars, 
the  average  distance  from  one  to  another  would  be  about 
25,000  times  the  distance  from  the  earth  to  the  sun.  At 
such  distances  gravitation  would  not  be  strong  enough  to 
bring  all  the  constituent  stars  under  the  influence  of  a  central 
force :  it  would  not  even  produce  velocities  of  interstellar 
orbital  motion  such  as  could  become  perceptible  to  our 
micrometric  instruments  during  the  comparatively  short 
period  since  precise  observations  were  commenced  by  ter- 
restrial man  (cf.  p.  322). 

Closely  related  to  the  clusters  are  the  nebulae  (p.  3). 
Indeed,  certain  clusters,  like  the  Pleiades,  are  so  completely 
interwoven  with  clouds  of  nebulous  matter,  and  with 
nebulous  lines  connecting  the  several  stars,  that  one  is  al- 
most inclined  to  regard  them  as  nebulae  partly  condensed  into 
stellar  nuclei.  But  the  true  nebulae  are  undoubtedly  gaseous  : 
spectroscopic  evidence  on  this  point  is  conclusive  (pp.  4,  283). 

" Planetary  nebulae"  are  a  class  of  nearly  circular  light- 
clouds  possessing  almost  planetary  central  disks.  Their 

352 


STARSHINE 

spectra  contain  certain  lines  belonging  to  nebulae  only, 
and  ascribed  to  incandescence  of  a  hypothetical  substance 
"nebulium."  At  their  centers,  nebulae  of  this  class  have 
at  times  nuclei  that  look  like  stars.  One  is  tempted  to 
imagine  them  in  the  last  stage  of  nebular  development,  and 
on  the  verge  of  becoming  starry.  There  exist  also  a  few 
ring-formed  nebulae. 

But  the  type-form  of  nebula  is  the  spiral  nebula  (cf. 
Plate  2,  p.  4).  Keeler  thought  there  are  120,000  objects 
of  this  form  within  the  photographic  range  of  his  big  reflect- 
ing telescope  at  the  Lick  observatory.  His  observations 
opened  the  eyes  of  astronomers  to  the  probability  that  the 
Laplacian  plan  of  cosmic  evolution  may  not  be  the  one 
generally  active;  that  a  single  sun  like  ours  is  less  likely 
to  occur  in  space  than  some  more  complicated  system  of 
suns,  resulting  perhaps  from  a  great  apparently  whirling 
complex  by  us  seen  as  a  spiral  nebula. 

The  Andromeda  nebula  is  by  far  the  largest  of  the  spirals  : 
for  modern  long-exposure  photographs  have  proved  its  spiral 
character,  though  it  was  always  supposed  to  be  elliptical 
or  ring-formed  until  celestial  photography  came  into  general 
use.  We  cannot  measure  its  distance  from  us  by  any  of 
our  parallax  methods  :  but  it  is  possible  to  fix  for  its  parallax 
a  limit  of  O."01.  The  parallax  cannot  be  much  larger  than 
this,  or  traces  of  it  would  reveal  themselves  to  our  instru- 
ments. But  assuming  this  parallax,  and  the  known  angular 
diameter  of  the  nebula  (one  and  a  half  degrees  of  arc),  it 
must  have  a  linear  diameter  540,000  times  as  great  as  the 
distance  between  the  earth  and  the  sun.1 

It  is  furthermore  of  interest  to  compare  this  nebula's 
possible  attractive  force  with  that  exerted  by  the  sun  on 

1  Note  44,  Appendix. 
2  A  353 


ASTRONOMY 

the  earth.  It  may  be  computed  that  if  the  nebular  density 
is  only  lorowooir  °^  ^ne  sun's>  ^ne  nebula  will  attract  the 
earth  as  much  as  the  sun  does.1  So  the  fact  that  we  find 
no  perturbative  attraction  whatever  in  the  solar  system 
resulting  from  the  nebulae,  proves  that,  though  enormous 
in  size,  they  are  of  an  almost  inconceivable  tenuity ;  in  fact, 
almost  without  any  density  whatever. 

To  the  foregoing  very  recent  researches  must  be  added  an 
observation  of  most  ancient  date,  but  one  that  all  the 
modern  theories  have  failed  to  explain  quite  fully.  This  is 
the  Milky  Way,  or  " galaxy,"  which  shows  itself  as  a  band 

of  small  stars  —  star 

/* 

dust  -  -  encircling 


the  sky  almost  like 
the    celestial    equa- 

FIG.  104.     The  Galaxy. 

tor,  ecliptic,  etc.     It 

has  many  starless  rifts  and  lanes,  and  several  "holes"  ;  not- 
ably the  "  coal-sack,"  situated  near  the  south  pole  of  the 
heavens.  It  contains  numerous  star-clusters,  but  few  nebulae. 
One  of  the  most  interesting  is  shown  in  Plate  29,  —  the ' '  North 
America"  nebula,  so  named  because  of  its  shape. 

The  galaxy,  resembling  a  great  circle  of  the  celestial  sphere, 
must  of  course  have  two  nodes,  or  points  of  intersection 
with  the  ecliptic  circle.  These  are  near  the  solstices,  and 
the  galactic  circle  makes  an  angle  of  about  60°  with  the  eclip- 
tic. According  to  Sir  William  Herschel,  the  stars  of  the  galaxy 
are  spread  out  in  a  thin  disk,  in  which  our  sun  is  also  situated. 
In  Fig.  104,  the  solar  system  is  shown  at  S,  and  the  galactic 
disk  is  within  the  rectangle.  Outside  the  rectangle,  the  stars 
are  fewer  and  farther  asunder.  It  is  evident  that  if  we  look 
in  the  direction  A  or  B  from  S,  we  shall  look  through  the 

1  Note  45,  Appendix. 
354 


PLATE  29.     The  North  America  Nebula. 


Photo  by  Barnard. 


STARSHINE 

thick  part  of  the  galaxy,  and  see  an  enormous  number  of 
stars  projected  on  the  celestial  sphere ;  but  if  we  look  toward 
(7,  we  shall  see  projected  on  the  sky  only  a  thin  part  of  the 
disk,  and  the  sparser  stars,  outside  it.  And  the  disk,  of 
course,  when  produced  outward  to  the  celestial  sphere,  will 
cut  out  the  galactic  great  circle. 

Actual  counts  of  stars  have  been  made  by  Herschel  and 
others,  to  ascertain  the  number  per  square  degree  of  sky 
surface  at  various  angular  distances  from  the  galaxy.  The 
numbers  are  found  to  vary  in  the  following  proportions : 

ANGULAR  DISTANCE  RELATIVE  NUMBER  OF  STARS 

FROM  GALAXY  PER  SQUARE  DEGREE 

90°  4 

60°  7 

30°  18 

0°  122 

The  unexplained  difficulty  with  Herschel's  explanation 
of  the  galaxy  still  remains.  The  extreme  minuteness  of 
the  galactic  stars  indicates  immense  distance,  as  does  also 
their  lack  of  observable  proper  motions  across  the  sky. 
But  if  so  enormously  distant,  how  can  the  galaxy  constitute 
a  single  disk-shaped  cluster  or  universe  ?  But  perhaps  we 
are  here  inventing  a  difficulty,  because  of  our  inability  even 
to  imagine  the  scale  of  the  sidereal  universe. 

However  this  may  be,  we  may  close  this  part  of  our  subject 
with  a  definite  statement  that  there  is  no  evidence  in  the 
possession  of  astronomers  to  indicate  a  "  central  sun  "around 
which  all  the  stars  are  circulating  in  their  orbits.  As  already 
stated,  we  now  believe  their  motions  more  nearly  resemble 
the  gyrations  of  the  molecules  in  a  gas  under  the  kinetic 
theory.  Gravitation  probably  takes  hold  in  an  appreciable 
degree,  only  when  a  couple  of  stellar  molecules  happen  to 
pass  near  each  other,  speaking  cosmically. 

355 


CHAPTER  XXI 

THE    UNIVERSE    ONCE    MORE 

THE  reader  may  recall  that  we  commenced  our  long  ex- 
planation of  astronomic  science  in  the  present  volume  with 
a  chapter  entitled  "The  Universe."  Now  that  we  at  last 
approach  the  end,  let  us  once  again  return  to  the  beginning, 
and  reexamine  the  evolutionary  processes  of  the  cosmic 
universe,  in  the  light  of  the  astronomic  knowledge  we  have 
been  able  to  gain. 

Cosmogony  is  a  name  given  to  the  various  theories  of  the 
universe  and  its  life-history  :  there  is  no  subject  more  entic- 
ing to  the  mind  of  man ;  none  in  which  he  is  more  prone 
to  be  misled  into  fields  of  mere  speculation,  quite  outside 
the  domain  of  strictest  logic,  based  on  irrefragible  observa- 
tional premises. 

We  have  already  mentioned  (p.  235)  the  Laplacian  nebular 
hypothesis,  with  its  rotating  nebulous  sun,  forming  planets 
by  the  successive  separation  of  rings  :  it  will  now  be  proper  to 
inquire  a  little  more  closely  into  the  admissibility  of  La- 
place's idea. 

It  will  be  well  to  begin  by  summarizing  the  known  facts 
that  are  favorable  to  Laplace  : 1 

1.  The  planetary  orbits  all  lie  nearly  in  the  same  plane; 
and  the  direction  of  orbital  motion  is  the  same  for  all  planets, 
and  for  the  sun's  axial  rotation. 

2.  The  orbits  are  all  nearly  circular. 

1  Laplace,  "Exposition  du  Systeme  du  Monde,"  p.  470,  in  Oeuvres  de 
Laplace,  Paris,  MDCCCXLVI. 

356 


THE   UNIVERSE   ONCE   MORE 

3.  With  the  exception  of  Uranus  and  Neptune,  the  equa- 
torial planes  of  the  planets,  and  even  the  orbital  planes 
of   their    satellites,    all    coincide    approximately   with    the 
fundamental  plane  of  all  the  orbits;    and  the  direction  of 
the  satellites'  orbital  motion  is  in  general  also  the  direction 
of  the  planets'  axial  rotation. 

4.  We   have   accepted    the  Helmholtz   theory   (p.  294), 
that  the  sun's  source  of  heat  must  be  sought  in  the  contrac- 
tion of  its  bulk :  in  that  case  the  sun  must  have  once  been 
incomparably  larger  than  at  present,  just  as  the  nebular 
hypothesis  requires.     Laplace,  of  course,  was  not  in  posses- 
sion of   Helmholtz'  calculations  when  his  own  theory  was 
published. 

Now,  as  a  matter  of  logic,  a  correct  theory  must  explain 
every  observed  fact  within  its  range.  A  single  contrary 
observation  may  destroy  logically  an  entire  theory,  no 
matter  how  many  other  observations  seem  to  confirm  it.  Let 
us  then  next  enumerate  some  of  the  objections  that  have 
been  advanced  against  the  nebular  hypothesis. 

1.  Phobos,  the  inner  satellite  of  Mars  (p.  222),  and  the 
inner  edge  of  Saturn's  ring  (p.  245),  revolve  in  their  orbits 
faster  than  the  axial  rotation  periods  of  Mars  and  Saturn 
respectively.     This   will   not   do :     the   contracting   planet 
should  rotate  faster  than  any  satellite  revolves  around  it, 
just  as  the  inner  planets  have  shorter  sidereal  periods  than 
the  outer  ones. 

2.  The  kinetic  theory  of  gases  would  lead  us  to  expect 
(pp.  167,  222)  that  a  gas  like  hydrogen  would  all  be  lost  into 
space  from  each  planet  in  the  form  of  molecules,  soon  after 
the  ring  was  thrown  off  from  the  sun,  on  account  of  the  very 
small  gravitational  pull  of  the  ring.     Yet  we  still  find  hy- 
drogen in  plenty  on  the  earth. 

357 


ASTRONOMY 

3.  The  throwing  off  of  the  rings  is  in  itself  an  hypothesis 
difficult    of    acceptance,    on    account    of    the    presumably 
extreme  rarity  of  the  outer  parts  of  the  solar  nebula,  and 
consequent  lack  of  cohesion.     And  why  was  the  process  of 
expelling  rings  intermittent  instead  of  continuous  ? 

4.  The  mechanical  movements  in  the  system  present  diffi- 
culties.    For  instance,  the  total  quantity  of  rotary  momen- 
tum now  belonging  to  Jupiter  is  about  -f^  of  the  total  belong- 
ing to  the  entire  solar  system  included  within  the  orbit 
of  Saturn.     Yet  Jupiter's  mass  is  only  about  one-thousandth 
of  the  total  mass  remaining  within  Saturn's  orbit,  including 
the  sun.     Are  we  then  to  suppose  that  the  present  sun,  at 
a  single  moment,  parted  with  so  small  a  fraction  of  its  mass, 
which  yet  carried  away  almost  all  its  rotary  power  ? 

Chamberlin  and  Moulton  have  endeavored  recently  to 
substitute  a  new  and  different  theory  for  that  of  Laplace. 
They  call  it  the  " planetesimal  hypothesis" ;  and  they  think 
the  recent  evolution  of  stellar  systems,  and  of  our  solar  system, 
began  with  the  accidental  close  approach  (not  collision)  of 
two  stars.  If  we  imagine  such  an  approach  to  have  taken 
place,  we  must  suppose  the  two  bodies  revolving  for  a  time 
in  orbits  around  their  common  center  of  gravity.  If  the 
approach  was  not  very  close,  these  orbits  would  be  open 
curves,  like  the  orbits  of  most  comets  when  they  approach 
the  sun  (p.  312).  The  two  bodies  would  separate  after  a 
certain  time,  and  would  never  again  pass  near  each  other. 

But  while  they  were  together,  the  gravitational  effect  of 
each  on  the  other  would  be  tremendous.  Doubtless  each 
would  eject  masses  of  highly  heated  gaseous  matter,  much  as 
the  great  red  hydrogen  prominences  (p.  293)  are  ejected  from 
our  sun.  Upon  these  ejected  gases,  and  upon  the  other  outer 
gaseous  envelopes  of  the  two  stars,  gigantic  tidal  forces 

358 


Spiral  Nebula,  seen  edgewise. 


Photo  by  Keeler. 


Owl  Nebula. 
PLATE  30.     Nebulae. 


Photo  by  Hale. 


THE   UNIVERSE   ONCE   MORE 

would  be  exerted.  Consequent  gigantic  tidal  effects  would 
be  produced  in  each  star  by  the  other.  Like  the  tidal 
crests  caused  by  the  moon  in  terrestrial  oceans  (p.  252), 
the  ejected  masses  would  travel  outwards  from  each  star, 
directly  toward  the  other  star,  and  directly  away  from  it. 

Afterwards,  these  masses  would  move  in  strange  orbits 
under  the  combined  gravitational  attractions  of  the  two 
stars :  we  can  in  a  way  trace  out  these  orbits  by  com- 
putational methods,  which  have  been  tried  by  Moulton. 
Chamberlin  calls  these  masses  "  planetesimals " ;  and  it 
is  assumed  that  they  make  their  appearance  in  great  num- 
bers and  at  short  intervals. 
Figure  105  shows  the  probable 
orbits  they  would  pursue.  The 
dotted  lines  indicate  various 
orbits ;  the  full  lines  show  the 

.     .  ,       ,  .  .  FIG.  105.     Planetesimal  Hypothesis. 

points  reached  at  a  given  in- 
stant of  time  by  the  several  planetesimals  pursuing  the 
dotted  orbits. 

When  we  look  at  the  result  of  such  a  performance,  what 
shall  we  expect  to  see?  If  we  assume  the  instant  of  time 
when  we  make  our  observation  to  be  that  instant  when  the 
several  planetesimals  have  just  reached  the  full  lines,  we 
shall  not  see  them  traveling  along  their  dotted  orbits,  but 
we  shall  see  them  all  lying  on  the  full  lines.  In  other  words, 
we  shall  see  a  spiral  nebula. 

Now  whatever  strength  there  may  be  in  this  hypothesis, 
there  can  be  no  doubt  that  Keeler's  observations  of  nebulae 
(p.  353)  establish  the  fact  that  the  spiral  is  the  normal  type 
of  nebula.  For  comparison  with  Plate  2,  p.  4,  we  give  in 
Plate  30  a  photograph  of  a  nebula  which  is  doubtless  another 
spiral,  but  seen  edgewise.  The  lower  part  of  the  plate 

359 


ASTRONOMY 

shows  the  "owl"  nebula  in  Ursa  Major,  which  looks  more 
like  a  Laplacian  nebulous  sphere.  Plate  31,  p.  360,  the 
"trifid"  nebula,  is  a  good  example  of  quite  irregular  shape. 

So  the  new  theory  gives  a  notion  as  to  the  possibility  of 
spirals  resulting  from  a  close  approach  of  two  stars,  which 
may  have  been  previously  moving  about  in  space  aimlessly, 
after  the  fashion  of  the  molecules  belonging  to  a  rarefied 
gas  (p.  347).  Thus  the  theory  goes  back  to  a  very  early 
stage  of  cosmic  development,  and  shows  how  stars,  even 
dark  ones,  can  be  transformed  into  nebulae. 

But  how  can  the  spiral  nebula,  in  turn,  develop  into  a  sun 
and  planets  such  as  we  have  in  our  solar  system?  Of  course, 
the  sun  is  simply  the  remaining  material  from  one  of  the 
original  stars  after  the  other  had  left  it.  If  the  approach 
was  near  enough  to  lead  to  a  permanent  orbital  proximity, 
we  might  perhaps  expect  a  binary  system.  And  the  sun 
provided,  it  is  easy  to  imagine  the  origin  of  the  planets. 
They  are  unusually  large  occasional  planetesimals,  increased 
gradually  by  the  accretion  of  other  smaller  ones,  swept  up 
by  them  as  they  moved  along  in  their  dotted  orbits.  When 
there  was  no  large  dominating  planetesimal  for  a  long  time, 
a  group  of  minor  planets  might  result.  The  satellites 
must  be  regarded  as  little  planetesimals  that  were  shot  out 
near  the  larger  ones  that  became  planets. 

Chamberlin  and  Moulton  have  traced  out  in  detail  the 
bearing  of  their  theory  upon  the  various  objections  that 
have  been  enumerated  against  the  Laplacian  idea,  and  with 
considerable  success.  The  great  advantage  of  their  hypoth- 
esis is  that  it  gives  us  an  origin  antedating  the  nebular 
stage ;  that  it  makes  a  cycle  of  cosmic  life  and  death ;  and 
especially  a  cycle  in  accord  with  the  actual  visible  appearance 
of  existing  nebulae.  This  the  Laplacian  theory  does  not  do. 

360 


PLATE  31.     The  Trifid  Nebula. 


Photo  by  Hale. 


THE   UNIVERSE   ONCE   MORE 

The  future  of  the  solar  system  seems  fairly  clear  under 
either  hypothesis.  The  present  state  of  affairs  is  one  of 
apparently  stable  equilibrium ;  and  should  continue,  unless 
an  accident  arrives  from  outside  the  system.  But  even 
without  such  accident,  the  solar  system  cannot  be  eternal, 
because  the  gradual  shrinkage  of  the  sun  cannot  continue 
forever.  When  the  time  comes  for  the  sun  to  lose  its  heat- 
radiating  power,  the  solar  system  must  become  cold  and 
dead.  If,  after  countless  ages,  it  shall  ever  thereafter  be 
revivified,  the  cause  will  be  a  fresh  approach  to  some 
other  star,  dark  or  brilliant,  whose  vast  disturbing  attraction 
will  once  more  break  up  the  solar  matter  into  a  mist :  and 
if  a  great  part  of  the  energy  remaining  in  the  system  shall 
be  transformed  into  heat,  then  that  mist  will  once  again  be  a 
fire-mist,  which  may  once  more  pass  through  all  the  stages 
of  cosmic  life  and  death. 


361 


APPENDIX 

THE  following  notes  contain  explanations  omitted  in  the  text, 
and  requiring  occasionally  a  knowledge  of  elementary  algebra, 
geometry,  and  trigonometry  as  far  as  the  solution  of  plane  right 
triangles. 

Note  i.     Declination  and  Right-ascension  (p.  35). 

Declination  corresponds  exactly  to  latitude  on  the  earth;  the 
declination  of  a  star  is  its  angular  distance  on  the  celestial  sphere 
measured  north  or  south  from  the  celestial  equator.  This  angular 
declination,  like  all  angles,  must,  of  course,  be  measured  on  some 
circle ;  for  measuring  it  we  must  imagine  a  circle  drawn  upon  the 
sky  from  the  star  to  the  equator,  and  perpendicular  to  the  equator. 
Such  a  circle,  drawn  for  the  purpose  of  measuring  declination, 
is  called  an  Hour-circle.  The  point  where  the  hour-circle  meets 
the  celestial  equator  may  be  called  the  foot  of  the  hour-circle.  The 
right-ascension  of  a  star  is  then  defined  as  the  angular  distance, 
measured  on  the  celestial  equator,  from  the  vernal  equinox  point 
on  the  equator  to  the  foot  of  the  hour-circle  drawn  from  the  star 
to  the  equator. 

Note  2.     Hour-angle,  etc.  (p.  37). 

We  may  now  define  also  the  term  "  hour-angle,"  which  is  closely 
related  to  the  hour-circle  used  in  measuring  right-ascension.  The 
hour-angle  of  a  star  is  defined  as  the  angular  distance,  measured 
like  right-ascension  on  the  celestial  equator,  from  the  meridian  to 
the  foot  of  the  hour-circle  drawn  from  the  star  to  the  equator. 
Thus  hour-angle  and  right-ascension  are  both  arcs  measured  on 
the  equator ;  both  arcs  have  one  end  in  common,  the  foot  of  the 
hour-circle;  but  the  other  ends  are  different,  being  respectively 
the  meridian  and  the  vernal  equinox. 

All  the  astronomical  terms  so  far  defined  are  exhibited  in  Fig. 
106.  It  represents  half  the  celestial  sphere,  the  half  which  is 

363 


ASTRONOMY 


above  the  horizon,  and  therefore  visible  to  us.  The  large  circle 
NESW  represents  the  horizon;  and  the  celestial  hemisphere  is 
shown  projected  down  upon  the  plane  of  this  horizon.  The  zenith, 
or  point  directly  overhead,  of  course  projects  down  into  the  center 
of  the  horizon  circle.  The  great  meridian  circle  appears  as  the  line 

NPZS,  since  it  must 
pass  through  the 
zenith  Z  as  well  as 
the  north  and  south 
points  of  the  horizon 
shown  at  N  and  S. 
The  celestial  north 
pole,  which  is,  by 
definition,  in  the  ce- 
lestial meridian,  will 
project  down  to 
some  point  P.  The 
celestial  equator, 
everywhere  90°  dis- 
tant from  the  pole 
P,  will  project  into 

FIG.  106.     The  Celestial  Sphere.  ,   J   , 

the    circle  WME. 

Any  star  selected  at  random  may  be  supposed  to  be  projected 
down  at  the  point  S'.  Then  S'D,  an  arc  drawn  on  the  sphere 
through  the  star  and  perpendicular  to  the  equator,  is  by  definition 
an  hour-circle.  It  is  evident  that  all  hour-circles  must  pass  through 
the  pole  P.  The  arc  DM  on  the  celestial  equator,  included  be- 
tween the  meridian  at  M  and  the  foot  of  the  hour-circle  at  D,  is 
the  hour-angle  of  the  star.  The  arc  S'A  is  the  star's  altitude,  or 
angular  elevation  above  the  horizon.  Finally,  if  we  draw  a 
short  piece  of  the  projected  ecliptic  circle,  we  may  take  V  to  be 
one  of  its  points  of  intersection  with  the  celestial  equator  WME, 
the  other  point  of  intersection  being  of  course  below  the  horizon. 
And  if  we  let  V  be  that  one  of  the  two  points  of  intersection  which 
we  have  called  the  vernal  equinox,  then  the  right-ascension  of  the 
star  S'  is  the  arc  VD,  measured  on  the  equator,  and  included 

364 


APPENDIX 


between  the  vernal  equinox  V  and  the  foot  of  the  hour-circle  at  D. 
The  arc  DS'  is  the  declination. 

The  above  astronomical  terms  may  be  divided  into  two  classes ; 
viz. :  those  that  retain  a  constant  position  among  the  stars  on  the 
celestial  sphere,  and  those  that  are  as  constantly  shifting  their 
positions  among  the  stars  on  account  of  the  daily  seeming  rotation 
of  the  whole  sphere.  Thus,  for  instance,  the  zenith,  which  is  the 
point  directly  overhead,  does  not  partake  of  the  seeming  turning 
of  the  sphere.  The  following  little  table  shows  the  two  classes  of 
terms : 


CHANGING  POSITIONS  AMONG  THE  STARS. 
Do  NOT  ROTATE  WITH  SPHERE 

Zenith 

Horizon 

Altitude 

Hour-angle 

Meridian 


UNCHANGING  POSITIONS  AMONG  THE  STARS. 
ROTATE  WITH  THE  SPHERE 

Celestial  Poles, 
Celestial  Equator 
Ecliptic  Circle 
All  hour-circles 
Right-ascension 
Declination 
The  stars,  sun,  etc. 
Vernal  Equinox 

Note  3.     Position  of  Celestial  Pole  (p.  40). 

The  relative  positions  of  the  celestial  pole  and  the  horizon  may 
be  made  clear  by  means  of  a  simple  diagram.  Figure  107  shows  a 
portion  of  the  earth,  with  its 
center  at  C,  and  the  observer 
on  the  surface  at  0.  The 
outer  concentric  circle  HPZE 
is  the  celestial  meridian,  on 
the  celestial  sphere.  The 
zenith  Z  will  be  directly  over 
the  observer  at  0,  on  the 


FIG.  107.     Position  of  Celestial  Pole. 


prolongation  of  the  observer's 
terrestrial    radius    CO.     The 

celestial  pole  must  be  at  some  point  of  the  celestial  meridian,  by 
definition.  Let  this  point  be  P.  The  celestial  equator  will  meet 
the  meridian  at  E,  90°  distant  from  P.  The  terrestrial  pole  will  be 
at  Pf,  and  Ef  will  be  a  point  of  the  terrestrial  equator.  The  angle 
E'CO  will  be  the  terrestrial  latitude  of  the  point  0,  since  it  is  the 

365 


ASTRONOMY 

angular  distance  of  the  point  0  from  the  terrestrial  equator  at  Ef. 
The  angle  PCH  is  the  altitude  or  angular  elevation  of  the  celestial 
pole  above  the  horizon  at  H.  For  H,  as  we  know,  is  the  north 
point  of  the  horizon  for  an  observer  at  0. 

But  %ZCE  =  %-PCH,  since  PC  is  perpendicular  to  CE,  and 
HC  perpendicular  to  ZC.  Hence  we  have  a  demonstration  that 
the  altitude  of  the  celestial  pole  is  everywhere  equal  to  the  terres- 
trial latitude  of  the  observer.  Thus,  as  stated  in  the  text,  this  alti- 
tude will  be  90°  to  an  observer  at  the  pole  of  the  earth,  where 
the  latitude  is  90°  ;  and  it  will  be  0°  to  an  observer  at  the  equator, 
where  the  terrestrial  latitude  is  likewise  0°. 

Note  4.     Stars  that  never  Set  (p.  43). 

It  is  evident  that  these  stars  are  the  ones  whose  diurnal  circles 
have  an  angular  distance  from  the  celestial  pole  less  than  PH, 
(Fig.  107) ;  i.e.  less  than  the  observer's  terrestrial  latitude. 
These  stars  will  have  a  declination  greater  than  (90°  —  latitude). 

Note  5.     Sidereal  Time  (p.  67). 

The  sidereal  time,  or  hour-angle  of  the  vernal  equinox,  is  the 
arc  VM  in  Fig.  106. 

To  make  the  definition  of  sidereal  time  perfectly  general,  as- 
tronomers count  all  hour-angles  westward  from  the  meridian,  and 
allow  them  to  increase  continuously  to  24h.  Thus,  an  hour- 
angle  lh  east  from  the  meridian,  corresponding  to  23h  sidereal 
time,  would  be  called  a  west  hour-angle  of  23b. 

Note  6.     Right-ascension  of  the  Meridian  (p.  67). 

Again  recurring  to  Fig.  106,  it  is  clear  from  our  definitions  that 
the  right-ascension  of  a  star  on  the  meridian  is  the  arc  VM ;  and 
we  have  seen  in  Note  5  that  this  identical  arc  is  also  the  sidereal 
time.  Therefore  the  sidereal  time  and  the  right-ascension  of  the 
meridian  at  any  instant  are  the  same. 

The  general  relation  of  hour-angle,  right-ascension,  and  sidereal 
time  may  also  be  deduced  from  Fig.  106.  We  have  from  our 
definitions : 

VD  =  right-ascension  of  star  Sr, 
DM  =  hour-angle  of  star  $', 
VM  =  sidereal  time. 
366 


APPENDIX 


And  since,  from  Fig.  106 : 

VM  =  VD  +  DM, 
it  follows  that  in  general : 

Sidereal  time  =  right-ascension  +  hour-angle. 

Hour-angle  =  sidereal  time  —  right-ascension. 
The  last  equation  enables  us  to  ascertain  the  hour-angle  of  a 
star  at  any  instant,  if  we  know  its  right-ascension,  and  have  a 
correct  sidereal  clock  at  hand. 

Note  7.     Hour-angle  of  Visible  Sun  (p.  68). 

In  Fig.  106,  if  we  let  Sr  be  the  visible  sun  at  any  instant,  its 
hour-angle  is  the  arc  DM,  measured  in  hours,  minutes,  and  seconds. 
This  same  arc  is  also  the  apparent  solar  time  at  that  instant.  .  " 

Note  8.     Terrestrial  and  Celestial  Meridians  (p.  73). 

If  we  imagine  a  line  drawn  from  the  center  of  the  earth  to  the 
observer,  and  thence  continued  outward  to  the  celestial  sphere, 
it  will  pierce  the 
sphere  at  the  ob- 
server's zenith.  The 
terrestrial  meridian, 
by  definition,  passes 
through  the  north 
pole  of  the  earth  and 
the  observer.  The 
celestial  meridian, 
also  by  definition, 
passes  through  the 
celestial  north  pole 
and  the  observer's 
zenith.  Therefore 
the  celestial  merid- 
ian is  a  projection 
of  the  terrestrial 
meridian  outward 
on  the  celestial  sphere.  Figure  108  is  like  Fig.  106,  with  the 
addition  of  a  second  celestial  meridian.  The  figure  represents  the 
celestial  sphere  projected  down  upon  the  horizon  of  New  York, 

367 


FIG.  108.     Time  Differences. 


ASTRONOMY 

of  which  the  zenith  appears  as  before  at  Z.  The  projection  of 
the  zenith  of  Greenwich  at  the  same  instant  is  at  Z' .  Therefore 
PZ'M'  will  be  the  projection  of  part  of  the  celestial  meridian 
of  Greenwich.  The  sun  and  vernal  equinox  are  projected  at 
Sf  and  V,  as  before.  Then  DM  is  the  sun's  hour-angle  at  New 
York;  DM',  its  hour-angle  at  the  same  instant  at  Greenwich. 
MM',  which  measures  the  angle  between  the  two  celestial  merid- 
ians, is  also  the  difference  of  the  two  hour-angles,  or  the  solar 
time  difference  between  New  York  and  Greenwich.  And  this  is 
the  same  as  the  longitude  difference,  measured  by  the  two  cor- 
responding terrestrial  meridians  on  the  earth  inside  the  celestial 
sphere. 

At  the  same  moment,  VM  and  VMf  are  the  hour-angles  of  the 
vernal  equinox  at  New  York  and  Greenwich ;  and  MM'  is  also 
the  sidereal  time  difference.  Consequently,  the  sidereal  and  solar 
time  differences  are  equal  and  identical ;  they  are  both  measured 
by  the  same  arc  MM'. 

Note  9.     Angle  of  Gnomon  (p.  79). 

It  is  evident  that  the  "factor"  in  the  table  is  simply  the  tangent 
of  the  latitude.  In  Fig.  24, 

be  =  ac  tan  bac, 

and  if  the  tabular  factor  is  tan  latitude,  the  construction  of  the 
figure  will  make  the  angle  bac  equal  to  the  latitude,  as  required 
for  the  gnomon. 

Note  10.     Mathematical  Principles  of  the  Sundial  (pp.  80,  84). 

To  demonstrate  the  correctness  of  the  construction  given  in  the 
text  for  drawing  a  sundial,  it  is  necessary  to  have  recourse  to 
the  well-known  formulas  of  spherical  trigonometry  relating  to  the 
solution  of  right-angled  spherical  triangles.  The  accompanying 
Fig.  109  represents  the  conditions  of  the  problem.  The  large 
circle  ZPNQS  is  the  celestial  meridian.  The  circle  NIVS  is  the 
horizon,  on  the  plane  of  which  the  dial  is  to  be  drawn.  The  center 
of  the  dial  is  at  0 ;  and  QP  is  the  axis  of  the  celestial  sphere.  As 
the  edge  of  the  gnomon  is  parallel  to  the  axis  QP,  we  may  regard 
it  as  lying  in  that  axis,  because  the  sun  will  appear  to  rotate  around 
the  edge  of  the  gnomon  (p.  84).  So  we  may  consider  the  edge  of 

368 


APPENDIX 


the  gnomon  to  start  at  0,  and  to  extend  a  short  distance  in  the 
direction  OP. 

Now  suppose  OIV,  situated  in  the  horizon  plane  NIVS,  to  be 
the  direction  in  which  the  shadow  falls  at  four  o'clock.  Then, 
remembering  that  solar  time  is  simply  the  hour-angle  of  the  sun, 
we  recall  that  "four 
o'clock"  means  that  the 

sun's    hour-angle    is    four  /  ./  \«?° 

hours,  or  60°.  We  may 
suppose  the  sun  to  appear 
at  the  point  S'  at  four 
o'clock.  Then,  from  the 
definition  of  hour-angle 
(p.  363),  the  sun  is  then 
distant  60°  from  the 
meridian;  or  the  angle 
ZPS'  is  60°.  The  op- 
posite angle  NPIV,  be- 
ing equal  to  ZPS',  is  thus 
also  60°. 

Now  let  us  consider  the  spherical  triangle  formed  on  the  celestial 
sphere  by  the  three  points  P,  N,  and  IV.  In  it  we  know  the  side 
PN,  for  it  is  the  elevation  of  the  celestial  pole  above  the  horizon, 
and  therefore  equal  to  the  latitude  of  the  place  where  the  dial  is 
to  be  used  (p.  365).  As  we  have  just  seen,  we  also  know  the  angle 
NPIV,  which  is  60°.  And  we  know  the  angle  PNIV  to  be  a 
right  angle,  because  the  celestial  meridian  is  perpendicular  to 
the  horizon. 

According  to  the  principles  of  spherical  trigonometry,  if  we 
know  one  side  and  one  acute  angle  of  a  right-angled  spherical 
triangle,  we  can  calculate  all  the  other  parts  of  the  triangle.  In 
the  present  problem,  we  need  only  calculate  the  side  NIV.  For 
this  measures  the  angle  NOIV,  which  is  the  dial  angle  for  the 
four-o'clock  line,  or  the  angle  which  the  four-o'clock  line  makes 
with  the  north-and-south  line  ON. 

In  the  same  way,  we  can  calculate  the  dial  angles  for  the  one- 
o'clock,  three-o'clock  lines,  etc.     The  twelve-o'clock  line,  or  noon- 
2s  369 


FIG.  109.     The  Sundial. 


ASTRONOMY 

line,  is  of  course  ON ;  for  at  noon  the  sun  is  on  the  meridian,  and 
the  shadow  of  a  gnomon  pointing  at  the  celestial  pole  will  then  fall 
due  north.  We  might  construct  the  dial  by  simply  laying  off  the 
proper  computed  angles  for  the  various  hours  from  the  dial 
center  0. 

The  trigonometric  formula  for  calculating   the  side  NIV,  or 
the  dial  angle  NO IV,  is : 

tan  NIV  =  tan  NPIV  sin  PN. 

And  if  we  let  : 

u  =  dial  angle  for  any  hour, 
t  =  corresponding  hour-angle  of  the  sun, 
I  =  latitude  of  the  place, 
then  the  general  formula  is : 

tan  u  =  tan  t  sin  I. 

The  dial  angles  calculated  by  this  formula  for  the  latitude  of 
New  York  are  as  follows : 


XII. 

I. 
II. 
III. 

IV. 
V. 
VI. 

0°     0' 
9°  56' 
20°  40' 
33°  10' 
48°  32' 
67°  42' 
90°    0' 

It  now  remains  to  show  that  the  construction  given  in  the  text 
(Fig.  25)  is  in  accord  with  the  above  general  formula.  In  this 
figure  we  have  really  drawn  two  half-dials,  so  as  to  allow  for  the 
thickness  of  the  gnomon.  To  prove  the  construction  of  Fig.  25 
correct,  we  have  now  to  show,  for  instance,  that : 

tan  cal  =  tan  15°  sin  I. 

The  factors  given  in  the  table  on  p.  80  are  the  sines  of  the 
latitudes  I.  Therefore,  since  we  made  Me  (Fig.  25)  equal  to  ca 
multiplied  by  the  factor  in  the  table,  it  follows  that : 

Me  =  ca  sin  I.  (1) 

We  made  the  angle  cMI  (Fig.  25)  equal  to  one-sixth  of  a  right 

370 


APPENDIX 


angle,  or  15°.     Therefore,  from  the  right-angled  plane  triangle 
Mel,  we  have : 

tan  15°, 


Me 
or: 

Ic  =  Me  tan  15°. 

Substituting  the  value  of  Me  from  equation  (1)  gives  : 

Ic  =  ca  tan  15°  sin  I, 
or: 

—  =  tan  15°  sin  I.  (2) 
ca 

Now  from  the  right-angled  plane  triangle  c/a,  we  have  : 

—  =  tan  cal.  (3) 
ca 

Substituting  from  equation  (2)  in  equation  (3),  we  have 

tan  cal  =  tan  15°  sin  I ; 

and  the  correctness  of  the  construction  in  Fig.  25  is  proved,  since 
the  above  equation  accords  with  the  general  form : 
tan  u  =  tan  t  sin  I. 

Note  ii.     Theory  of  Foucault  Experiment  (p.  91). 

We  have  explained  the  conditions  of  the  problem  if  the  experi- 
ment were  performed  at  the  north  pole  of  the  earth.  There  the 
point  of  suspension  of  the  pendulum's  wire  would  of  course  be 
situated  in  the  prolongation  of  the  earth's  axis,  and  would  conse- 
quently not  move  as  a  result  of  the  earth's  axial  rotation,  which 
is  the  only  motion  of  the  earth  here  requiring  consideration.  In 
any  other  latitude,  the  point  of  suspension  would  go  around  as 
the  earth  rotates :  it  is  therefore  necessary  to  explain  further  the 
statement  that  the  direction  in  space  of  the  pendulum's  plane  of 
oscillation  tends  to  remain  constant.  The  fact  is  that  when  the 
point  of  suspension  moves,  the  plane  of  oscillation  moves  also ; 
but  it  tends  to  occupy  a  position  constantly  parallel  to  itself. 
Any  one  can  satisfy  himself  that  this  is  correct  by  fastening  a 
small  metal  ball  to  a  string  and  letting  it  oscillate,  the  end  of  the 
string  being  held  in  the  experimenter's  hand.  It  will  be  found 
that  the  experimenter  may  walk  across  the  room,  carrying  the  end 

371 


ASTRONOMY 


of  the  string ;  yet  the  plane  of  oscillation  will  remain  constantly 

parallel  to  itself. 

So  much  being  premised, we  may  now  proceed  to  calculate  the 

rate  at  which  the  marks  under  the  pendulum  should  rotate.  Let  us 
suppose  we  start  the  pendulum  swinging  in 
a  north-and-south  direction,  and  therefore 
directly  under  the  celestial  meridian,  and  in 
the  plane  of  the  meridian.  In  Fig.  110,  let 
NABCS  be  the  meridian  directly  over  the 
pendulum  when  we  start  it  swinging,  and 
suppose  it  swings  between  two  points  in  the 
room  corresponding  to  the  points  A  and  B 
of  this  meridian.  In  a  second  of  time  the 
earth's  rotation  will  have  brought  a  new 
celestial  meridian  over  the  swinging  pendu- 
lum, and  the  old  one  will  have  gone  to  the 
position  NA'B'C'S. 

But  the  pendulum  will  still  swing  parallel 
to  the  plane  of  the  first  meridian,  and  the 
rotation  shown  by  the  experiment  will  be 
equal  to  the  angle  between  the  two  meridians. 

Let  us  draw  Fig.  Ill  to  show  this  angle.     This  figure  is  like  a 

map  in  a  geography  book.     If  the  original  meridian  was  AB,  and 

the  meridian  at  the  end  of  one  second  A'Bf,  the  rotation  shown 

by  the  pendulum  will  be  the  angle  between  these  two  lines.     If 

we  draw  A'M  perpendicular  to  BB',  the  rotation 

angle  will  be  the  angle  MA'B1 '.     Let  us  call  this 

angle  a. 

It  is  well  known  that  on  a  map  of  the  earth's 

equatorial  regions  the  terrestrial  meridians  are 

practically  parallel :  there  is  no  "  convergence  of 

meridians"  there  ;    and  there  would  be  no  Fou- 

cault  effect.     Near  the  pole  the  angle  between 

the  meridians  is  a  maximum  :  there  the  Foucault 

effect  is  also  greatest. 

In  this  way  we  translate  our  astronomical  problem  into  terms  of 

geometry ;   it  is  now  merely  a  question  of  simple  geometry  to  as- 

372 


s 

FIG.  110.    The  Foucault 
Experiment. 


BM     B 

FIG.  111.    The  Fou- 
cault Experiment. 


APPENDIX 

certain  the  angle  of  convergence  between  two  neighboring  meridians 
on  the  earth  in  any  latitude,  such  as  that  of  New  York,  for  instance  ; 
and  this  angle  will  be  the  Foucault  pendulum  rate  of  rotation. 

We  see  at  once  from  Fig.  Ill  that,  in  any  latitude,  we  have 
from  the  triangle  A'B'M: 


and  since  for  very  small  angles  like  a  the  tangent  and  the  arc  are 
equal,  we  may  write  : 

« 


Referring  again  to  Fig.  110,  which  we  may  now  take  to  represent 
the  earth  instead  of  the  celestial  sphere,  we  observe  that  the  latitude 
arcs  A  A',  BB',  and  CC'  are  all  arcs  of  circles  whose  radii  are 
PA',  P'B',  and  OC'.  The  last  radius  OC'  is  the  earth's  radius, 
because  we  shall  consider  CC'  to  be  an  arc  of  the  equator.  Now 
suppose  the  point  B'  to  correspond  to  the  terrestrial  latitude  V  . 
Then  V  is  the  angle  B'OC',  for  the  latitude  is  the  angular  distance 
of  Bf  from  the  equator.  But  the  line  P'B'  in  the  plane  of  the  circle 
NA'B'S  is  proportional  to  the  cosine  of  the  angle  B'OC'.  Simi- 
larly, the  radii  of  all  arcs  like  A  A',  BB',  etc.,  are  simply  proportional 
to  the  cosines  of  the  latitudes  corresponding  to  the  points  A',  B'} 
etc. 

But  the  arcs  themselves  must  be  proportional  to  their  radii. 
So  it  follows  that  the  linear  lengths  of  the  arc  A  A',  BB',  are  also 
proportional  to  the  cosines  of  the  corresponding  latitudes. 

We  have  called  I'  the  latitude  corresponding  to  the  point  B'. 
Let  us  call  I  the  latitude  corresponding  to  A'.  Now  we  have  found 
that  the  arcs  A  A'  and  BB'  are  proportional  in  length  to  the 
cosines  of  the  latitudes  I  and  V  .  Therefore  the  difference  between 
A  A'  and  BB'  must  be  proportional  to  the  difference  of  the  same 
cosines,  which  we  may  express  by  the  following  equation,  in  which 
K  is  simply  a  constant  denoting  proportionality  : 

BB'  -  AA'  =  K  (cos  V  -  cos  Z). 

But,  from  Fig.  Ill  : 

MB'  =  BB'  -  AA'. 
373 


ASTRONOMY 

Consequently,  from  the  preceding  equation  : 

MB'  =  K  (cos  V  -  cos  Z).  (2) 

Now,  in  Fig.  110,  draw  the  line  A'Q  perpendicular  to  P'B',  com- 
pleting a  little  right-angled  triangle  A'B'Q.  (We  may  regard  the 
short  arc  A'B'  as  here  equivalent  to  a  straight  line.)  Then  we 
have : 

QB'  =  cos  I1  -  cos  Z, 

and  :  QB'  =  A'B'  sin  QA'B'. 

But:  QA'B'  =  B'OC'  =  V  \ 

therefore:  QB'  =  A'B'sml'. 

But:  A'B'  -  (Z-  V). 

Consequently :  QB'  =  cos  V  -  cos  I  =  (Z  -  Z')  sin  Z'. 

It  then  follows  from  equation  (2)  that : 

MB'  =  K(l  -  I')  sin  V.  (3) 

We  also  have,  obviously,  from  Fig.  Ill : 

MA'   =1-1'.  (4) 

Now  substituting  from  equations  (3)  and  (4)  in  equation  (1),  we 
have  finally : 

«  =  K  sin  I'.  (5) 

This  simple  equation  (5)  establishes  the  important  principle 
that  the  rate  of  rotation  of  the  Foucault  pendulum  in  one  second 
must  everywhere  be  proportional  to  the  sine  of  the  latitude  of  the 
place  where  the  experiment  is  performed. 

It  is  further  obviously  indifferent  whether  the  original  impulse 
was  given  to  the  pendulum  in  the  direction  of  the  meridian ;  for 
whatever  angle  the  original  impulse  made  with  the  original  merid- 
ian, at  the  end  of  one  second  of  time  that  angle  will  have  changed 
by  the  same  quantity  «  with  respect  to  the  meridian. 

It  is  quite  easy  to  find  the  value  of  the  constant  K  in  equation 
(5).  For  at  the  north  pole,  sin  I'  =  1,  since  V  =  90°.  Therefore, 
at  the  pole,  equation  (5)  becomes  : 

a  =  K. 

But  we  already  know  that  at  the  pole  the  pendulum  must 
make  one  complete  revolution  of  360°  in  24  hours.  So  it  must 

374 


APPENDIX 

there  revolve  at  the  rate  of  15°  per  hour,  or  15'  per  minute.     With 
this  value  of  K  we  therefore  have  in  any  latitude  V  : 

Rate  of  revolution  =  (15'  per  minute)  sin  V. 
In  New  York,  for  instance, 

V  =  40°  48',  sin  V  =  0.65. 
Rate  of  rotation  =  9/75  per  minute. 

The  above  demonstration  of  the  Foucault  pendulum  theory  is 
not  rigorous,  but  it  is  sufficiently  accurate  for  ordinary  purposes, 
provided  the  duration  of  the  experiment  is  not  much  greater  than 
one  hour. 

Note  12.     The  Torsion  Constant  (p.  108). 

The  problem  of  ascertaining  the  torsion  constant  T  from  the 
time  of  oscillation  of  the  torsion  balance  is  quite  analogous  to  the 
corresponding  problem  of  determining  the  length  of  an  ordinary 
pendulum  from  its  time  of  vibration.  It  is  shown  in  books  on 
physics  that  if  we  let  : 

t  =  vibration  time  of  an  ordinary  simple  pendulum, 
I  =  length  of  the  pendulum, 
g  =  the  force  of  gravity  on  the  earth, 
*  =  the  ratio  3.1416, 
then  : 


An  analogous  formula  exists  in  the  case  of  the  torsion  balance, 
except  that  instead  of  g,  the  force  of  gravity,  the  formula  involves 
T,  the  torsion  constant.  We  now  let  I  represent  the  entire  length 
ab  of  the  torsion  balance  arm  (Figs.  32  and  33),  and  m  the  mass  of 
either  small  ball  a  or  b.  Then  the  torsion  balance  formula  is  : 


and  those  readers  who  are  acquainted  with  the  science  of  mechanics 

will  not 

balance. 


will  note  that  2m-\   is  the  "moment  of  inertia"  of  the  entire 


375 


ASTRONOMY 

Solving  this  equation  for  T  gives  : 


and  this  equation  will  make  known  the  value  of  T  for  any  torsion 
balance  after  we  have  observed  its  vibration  time  t,  measured 
the  length  of  the  arm  ab,  and  ascertained  m  by  weighing  the  small 
balls  in  an  ordinary  balance. 

Note  13.     The  Cavendish  Experiment  (p.  110). 

Returning  now  to  Fig.  33,  let  us  use  the  following  notation  : 
M  =  mass  in  grams  of  either  big  lead  ball, 
m  =  mass  in  grams  of  either  small  ball, 
d  =  measured  distance  in  centimeters  from  the  position  of  rest  b' 

to  Bf,  the  center  of  the  big  lead  ball, 
g  =  the  acceleration  due  to  the  "force  of  gravity,"  as  used  in 

physics,  or  981  centimeters, 
I  =  length  of  torsion  balance  arm,  or  the  distance  ab,  in  centi- 

meters, 

/  =  the  force  with  which  both  big  balls  turn  the  balance. 
Now,  according  to  Newton's  law,  the  attractive  force  between 
the  balls  B'  and  V  is  (p.  103)  : 

G^j±,  (1) 

in  which  formula  G  is  a  so-called  "  gravitational  constant,"  in- 
troduced to  indicate  that  the  attraction  is  proportional  to——' 

not  equal  to  it. 

The  distance  from  Bf  to  a'  is  : 


consequently,  the  attractive  force  existing  between  B'  and  a'  is  : 


Both  forces  (1)  and  (2)  tend  to  turn  the  torsion  balance.  They 
act  against  each  other,  however,  tending  to  rotate  the  balance 
in  opposite  directions.  And  the  force  (1)  is  larger  than  (2)  ; 
so  that  it  will  determine  the  final  direction  of  rotation. 

376 


APPENDIX 

Furthermore,  the  entire  force  (1)  tends  to  turn  the  balance, 
while  only  a  small  part  or  " component"  of  (2)  has  such  an  effect. 
We  can  easily  find  this  component,  which  acts  from  B'  upon  a'  so 
as  to  turn  the  balance.  According  to  the  so-called  "  parallelogram 
of  forces"  this  component  is  : 


^  ^sinB'o'6', 

or : 

Q    Mm  d 

d 
or,  finally : 


Mmd  .*  (3) 


The  effective  force  tending  to  rotate  the  balance,  and  resulting 
from  the  big  ball  B' ',  will  be  the  difference  of  (1)  and  (3).  It  will 
be  J/,  since,  in  our  notation,  /  is  the  force  with  which  both  big  balls 
tend  to  rotate  the  balance.  By  subtracting  (3)  from  (1)  we  thus 
obtain  the  equation : 

%f  =  GMm(~2-     — -\  (4) 

For  brevity,  let  us  put : 


Then  we  have  : 

if  =  GMmD,  (6) 

and,  solving  for  M,  we  obtain  : 


The  force  /  must  be  determined  from  observations  of  the  torsion 
balance,  when  under  the  influence  of  the  big  lead  balls.  Trans- 
ferring these  big  balls  from  the  position  A',  B'  to  the  position  A", 
B"  usually  rotates  the  balance  through  a  very  small  angle  only. 
It  is  therefore  necessary  to  measure  this  angle  by  very  delicate 
means.  For  this  purpose  a  small  light  mirror  is  attached  to  the 
center  of  the  arm  ab  of  the  balance,  and  rotation  is  measured  by 
allowing  a  strong  beam  of  light  to  fall  on  this  mirror,  and  to  be 

377 


ASTRONOMY 

thence  reflected  upon  a  scale  at  some  distance  from  the  apparatus. 
The  rotation  of  the  balance  is  thus  magnified,  and  can  be  measured 
without  difficulty. 

To  introduce  these  measures  into  our  formulas,  let  : 
«  =  the  total  change  in  centimeters  of  the  light  on  the  scale  brought 

about  by  changing  the  big  balls  from  A',  B'  to  A",  B". 
Q  =  the  distance  of  the  scale  from  the  mirror. 

To  employ  the  units  usual  in  measures  of  this  kind,  we  must 
reduce  the  motion  a  to  what  it  would  have  been  on  a  scale  at 

unit  distance  from  the  mirror.     This  would  be  ^-    We  must  also 

allow  for  the  well-known  fact  that  a  moving  reflected  beam  changes 
its  direction  twice  as  fast  as  the  mirror  turns.  This  reduces  the 

motion  on  the  scale  at  unit  distance  to  —  —  •     Finally,  we  must 

again  divide  by  2  to  obtain  the  effect  corresponding  to  the  half 
motion  W,  instead  of  the  whole  motion  b"b',  since  we  are  calcu- 
lating the  disturbance  of  the  balance  from  a  position  of  rest,  and 
have  measured  its  motion  between  two  positions  of  extreme 
disturbance.  This  gives  the  observed  motion  on  the  scale,  to  be 
used  in  the  further  calculations,  as  : 

Tn  (8) 

4Q 

Now  it  is  a  principle  underlying  the  torsion  or  twisting  of  rods 
or  fibers,  a  principle  verified  easily  by  experiment,  that  the  force 
required  to  twist  the  rod  or  fiber  through  any  angle  is  proportional 
to  that  angle.  For  instance,  if  a  certain  force  would  turn  the 
torsion  balance  through  an  angle  of  10°,  it  would  require  just 
twice  as  much  force  to  turn  it  through  20°.  It  follows  from 
this  principle,  and  from  the  definition  of  the  torsional  constant  T, 
that  the  force  required  to  turn  the  balance  through  the  angle  (8) 


where  readers  familiar  with  Mechanics  will  note  that  T  is  really 
the  ''turning  moment"  for  unit  angle  applied  at  unit  distance 
from  the  center. 

378 


APPENDIX 

This  expression  (9)  is  not  yet  equal  to  the  force  /,  because  /  is 
applied  at  the  ends  of  the  balance  arms  where  the  small  balls  are. 

The  length  of  this  balance  arm  being  -,  we  see  that  (9)  must  be 

2 

equal  to  /  -  ;  and  so  we  may  write  the  equation : 

±T-fL  .  (10) 

From  this  we  have : 

/(observed)  =          -  (11) 


We  have  already  obtained  in  Note  12  (p.  375)  an  expression  for 
T  as  follows : 

-.2™  72 
m   __   "    f i  to  /-J  n\ 

o  /2 

With  the  help  of  equations  (11)  and  (12)  we  can  compute  the 
observed  force  /  from  our  observations  of  «  and  Q,  and  the  known 
dimensions,  etc.,  of  the  parts  of  the  balance. 

Next  we  can  establish  easily  an  expression  for  the  attractive 
force  existing  between  either  little  ball  and  the  earth.  For  this 
purpose,  let 

R  =  the  radius  of  the  earth,  in  centimeters, 
E  =  mass  of  the  earth,  in  grams. 
Then  we  have : 

Attractive  force  between  small  ball  and  earth  =  G-—^'       (13) 

Equation  (13)  follows  directly  from  Newton's  law,  if  we  recall 
that  the  earth  attracts  bodies  exterior  to  it  precisely  as  if  the  entire 
mass  of  the  earth  were  concentrated  at  its  center.  Thus  the  radius 
of  the  earth  becomes  the  distance  between  the  earth  and  the 
small  ball,  and  its  square  appears  in  the  denominator  of  equa- 
tion (13). 

Furthermore,  according  to  the  teaching  of  Physics,  the  attractive 
force  existing  between  the  small  ball  and  the  earth  is  also  measured 
by  the  weight  of  the  small  ball,  since  weight  is  merely  the  result 
of  such  attractive  force  of  the  earth.  And  in  physics^  the  weight 

379 


ASTRONOMY 

of  any  object  is  shown  to  be  equal  to  its  mass  multiplied  by  the 
force  of  gravity,  g.     So  we  have  : 

Attractive  force  between  small  ball  and  earth  =  mg.         (14) 
Equating  the  right-hand  members  of  equations  (13)  and  (14) 

gives  :  „ 

nEm  /1  Kv 

mg  =  G—;  (15) 

or  :  P2 

E  =^f  (16) 

Cr 

If  we  now  divide  equation  (16)  by  equation  (7),  we  obtain  : 

(17) 


We  now  obtain  the  value  of  /  from  equation  (11)  by  the  help  of 
equation  (12).     This  gives: 

,  _ 

= 


Substituting  from  equation  (18)  in  equation  (17)  gives,  finally  : 

(19) 

Equation  (19)  enables  us  to  calculate  the  mass  of  the  earth,  E,  in 
terms  of  the  mass  of  either  big  lead  ball,  M.  It  will  be  noted  that 
the  only  quantities  used  in  equation  (19)  and  actually  observed  in 
the  Cavendish  experiment  are  «  and  Q.  Most  of  the  other  quanti- 
ties are  ascertained  by  measurements  and  weighings  before  the  tor- 
sion balance  is  put  together.  The  time  of  vibration,  t,  is  found  in 
seconds  by  observing  the  combined  duration  of  a  considerable  num- 
ber of  oscillations,  made  with  the  big  lead  balls  entirely  removed. 

In  the  actual  apparatus  mounted  for  use  in  the  astronomical 
lecture-room  at  Columbia  University,  New  York,  the  following 
numerical  data  exist  : 

t  =  281.5  seconds, 
d  =  5.3  centimeters, 
/  =  3.6  centimeters, 
g  =  981  centimeters  j 
TT  =  3.1416  centimeters, 
M  =  2750  grams, 
380 


APPENDIX 

and  the  radius  of  the  earth  is  : 

R  =  6.371  X  108  centimeters. 
With  these  numbers  we  obtain  from  equation  (19) : 

E  =  0.30  X  1027  -  2  grams. 
« 

In  an  actual  experiment  the  writer  found  : 
Q  =  189  centimeters, 
«  =  10.86  centimeters. 
Therefore : 

Q  =  17.4, 
and : 

E  =  5.22  X  1027  grams. 

The  present  accepted  value  of  the  earth's  mass  is : 

6  X  1027  grams ; 

so  that  the  result  of  the  above  lecture-room  experiment  is  fairly 
satisfactory. 

Note  14.     Linear  Distances  from  Angles  Alone  (p.  119). 

The  simple  Figure  112  shows  the  correctness  of  the  principle 
stated  in  the  text.  Suppose,  for  instance,  that  the  three  angles 
of  a  triangle  are  given,  and  it  is  required  to 
draw  the  triangle.  It  is  not  possible  to  do  so  ; 
because,  with  the  given  angles,  we  do  not 
know  whether  we  should  make  it  of  the 

,  i         •        7->  .,  m       FIG.  112.     Distance  from 

size  A,  or  the  size  B,  or  any  other  size.     lo  Angles 

know  the  triangle  fully,  we  must  know  the 
length  of  at  least  one  side.     Angles  alone  enable  us  to  draw  a  figure 
which  is  geometrically  similar  to  the  required  figure,  but  they  do 
not  enable  us  to  draw  the  figure  itself  to  scale. 

Note  15.     Calendar  Rule  (p.  144). 

To  demonstrate  this  rule,  we  begin  by  assuming  that  our  era 
commenced  with  a  year  numbered  0,  so  that  1913  was  the  1914th 
year  of  the  era.  Of  course  there  was  not  really  an  initial  year  0, 
but  we  can  imagine  the  calendar  extended  to  that  time.  Then  the 
principle  on  which  our  rule  is  based  consists  in  calculating  the 
number  of  days  from  January  1  of  the  year  0  to  the  date  under  in- 

381 


ASTRONOMY 

vestigation,  and  ascertaining  how  many  weeks  elapsed  in  the 
interval. 

It  happens  that  January  1  of  the  year  0  was  a  Sunday.  Let  us 
next  compute  the  number  of  days  between  January  1  of  the  year  0 
and  March  1  of  any  year,  such  as  1913.  We  select  March  1, 
because  it  is  desirable  for  the  moment  to  use  a  date  that  follows 
the  possible  extra  day  inserted  as  February  29  in  leap-years. 

Let  us  indicate  the  year  number,  such  as  1913,  by  the  letter  y, 
and  the  century  number,  such  as  19  in  the  year  1913,  by  the  letter 
c.  The  total  number  of  days  from  January  1  in  the  year  0,  to 
March  1  of  the  same  year,  is  59,  for  the  year  0  was  not  a  leap-year. 
Consequently,  if  there  were  no  leap-years,  we  should  have : 

No.  of  days  from  Jan.  1,  year  0,  to  Mar.  1,  year  y  =  365  y  +  59. 

As  each  leap-year  adds  one  day,  we  must  increase  this  by  the 
number  of  leap-years  from  the  year  0  to  the  year  y}  and  including 
the  year  y,  if  it  be  a  leap-year.  To  find  this  number,  let  us  divide 
c  and  y  by  4,  and  call  the  remainders  after  the  division  n  and  r3. 
Then  it  is  clear  that  under  the  Gregorian  rule  the  number  of  leap- 
years  will  be : 

i  (y  -  r,)  -  c  +  \  (c  -  n). 

Furthermore,  this  number  will  be  a  whole  number,  because  we  can 
prove  easily  that  y  —  r3  and  c  —  r\  are  both  divisible  exactly  by  4, 
without  remainder. 

The  proof  of  this  is  as  follows  :  If  we  divide  any  number  what- 
ever, N,  by  some  other  number  Z>,  and  find  from  the  division  a 
quotient  Q  and  a  remainder  R ;  then,  if  we  divide  N  —  R  by  D,  we 
shall  again  find  the  same  quotient  Q,  but  the  remainder  will  now 
be  0.  Thus,  if  we  divide  1913  by  4,  we  find  the  quotient  Q  is  478 
and  the  remainder  R  is  1.  If  we  now  subtract  this  remainder  1 
from  the  original  number  1913,  we  have  for  N  —  R,  1912.  This 
being  divided  by  the  same  divisor  4,  gives  the  old  quotient  Q  as 
478,  but  the  remainder  is  now  0.  This  shows  that  our  expression 
for  the  number  of  leap-years  is  a  whole  number,  as  it  should  be. 

We  then  have,  by  addition  of  the  number  of  leap-years : 

Total  no.  of  days  from  Jan.  1,  year  0,  to  Mar.  1,  year  y 
-  365  y  +  59  +  J  (y  -  rg)  -  c  +  i  (c  -  n). 
382 


APPENDIX 

Now,  if  March  1  in  the  year  y  is  a  Sunday,  like  the  first  day  of 
the  era,  the  above  number  must  be  divisible  exactly  by  7.  But  if 
March  1  in  the  year  y  is  Monday,  one  day  later  than  Sunday, 
we  can  make  the  above  number  divisible  by  7  if  we  subtract  1 
from  it;  and  1  is  the  week-day  number  for  Monday,  minus  1. 
Similarly,  for  Wednesday,  for  which  the  week-day  number  is  4,  we 
would  subtract  3.  In  general,  let  us  indicate  by  w  the  week-day 
number  of  March  1,  whatever  it  may  be  in  the  year  y,  and  subtract 
w  —  1  from  the  above  total  number  of  days.  This  gives  : 

365  y  +  59  +  I  (y  -  r,)  -  c  +  J  (c  -  n)  -  (w  -  1), 

and  this  number  is  now  always  divisible  exactly  by  7. 

Our  real  problem  is  to  determine  (w  —  1)  from  the  fact  that  the 
number  just  obtained  is  thus  divisible  exactly  by  7.  In  doing  this 
we  may  evidently  increase  or  diminish  our  number  by  any  exact 
multiple  of  7  without  impairing  its  divisibility  by  7  or  affecting  the 
value  of  w.  We  shall  introduce  two  new  remainders  r2  and  r4,  by 
dividing  the  century  number  c,  and  the  year  number  y,  by  7,  just 
as  we  have  already  divided  them  both  by  4. 

This  having  been  done,  we  may  correct  our  total  number  of 
days  as  follows,  noting,  of  course,  that  each  number  added  or  sub- 
tracted is  divisible  exactly  by  7.  We  shall  add  : 


subtract  364  y  +  56, 

add  7  TI  +  7  r3, 

subtract  3  (y  —  r4)  +  (c  —  r2), 

and  so  our  total  number  becomes  : 

3  +  r2  +  5  r3  +  5  n  +  3  r4  -  (w  -  1). 

This  number  is  now  made  up  of  remainders  only.  It  will  be  a 
comparatively  small  number,  as  no  remainder  is  larger  than  6  ; 
and  it  is  still  divisible  exactly  by  7.  It  is  therefore  clear  that 
(w  —  l)  is  simply  the  remainder  that  will  occur  in  the  division  by 
7  of  the  number  : 

3  +  5  n  +  r2  +  5  7-3  +  3  r4, 

and  thus  (IP  —  1)  is  determined  for  March  1  in  the  year  y. 

383 


ASTRONOMY 

But  we  need  to  find  (w  —  1)  for  any  day  in  the  year  y,  not  merely 
for  March  1.  To  accomplish  this  for  any  other  day  in  March, 
say  the  3d,  for  instance,  we  have  merely  to  add  2  to  the  above 
number,  before  dividing  by  7,  because  March  3  comes  two  days 
later  than  March  1.  In  general,  if  we  indicate  by  d  the  date  in 
March  for  which  the  week-day  is  required,  we  must  add  (d  —  1) 
to  the  above  number.  This  gives,  for  March  dth  : 

3  +  5  ri  +  r2  +  5  r3  +  3  r4  +  (d  -  1), 
or  :  2  +  5  ri  +  r2  +  5  r3  -f-  3  r4  +  d ; 

and  this  number  being  divided  by  7  will  give  the  (w  —  1)  of  March 
d  for  a  remainder. 

The  same  expression  will  hold  for  April,  if  we  add  31,  because 
there  are  31  days  in  March.  Adding  31,  and  deducting  28,  an  exact 
multiple  of  7,  gives  for  April : 

5  +  5  ri  +  r2  +  5  r3  +  3  r4  +  d. 

A  similar  expression  holds  for  each  month,  a  difference  occurring 
only  in  the  number  at  the  beginning  of  the  expression.  If  we  in- 
dicate that  month-number  by  m,  we  may  write  for  any  month  : 

in  +  5  7*1  +  r2  +  5  r3  +  3  r4  +  d. 

The  values  of  m  for  the  various  months  may  then  be  written  in  a 
little  table  (see  Rule,  p.  144).  In  forming  this  table  it  is  necessary 
to  remember  that  there  will  be  a  slight  difference  between  the 
ra's  for  leap-years  and  ordinary  years.  We  have  started  with 
the  formula  for  March  1,  in  order  to  avoid  this  difference  as 
much  as  possible.  After  that  date  in  the  year  there  is  no  difference. 
But  in  January  and  February  the  leap-year  m's  are  smaller  by 
1  than  those  for  ordinary  years,  on  account  of  the  interpolated 
February  29. 

The  entire  rule  may  be  arranged  in  the  accompanying  tabular 
form.  That  part  of  the  formula  which  does  not  vary  in  a  whole 
century,  namely,  5  r\  +  r^  we  have  designated  by  K.  In  the  Julian 
calendar  K  is  evidently  always  0,  because  there  is  no  century  ex- 
ception in  the  leap-year  rule  of  that  calendar.  For  the  sake  of 
symmetry,  we  have  here  indicated  the  final  remainder  (w  —  1) 
by  r8. 

384 


APPENDIX 

CALCULATION  OF  WEEK-DAY,  GREGORIAN  OR  JULIAN  CALENDAR 


FORMULA 

TABLE  OF  m 

(d=Day  of  the  Month) 

Divide 

by 

and  call 
the 
remainder 

Ord'y 
Year 

Leap 
Year 

WEEK-DAY  Nos. 

Century  No. 

4 

Ti 

Jan. 

6 

5 

1 

Sunday 

Century  No. 

7 

r* 

Feb. 

2 

1 

2 

Monday 

Year  No. 

4 

rs 

March 

2 

2 

3 

Tuesday 

Year  No. 

7 

n 

April 

5 

5 

4 

Wednesday 

5r,  +  3r«-f-in 

71 

May 

0 

0 

5 

Thursday 

+  m  +  d\ 

. 

r$ 

June 

3 

3 

6 

Friday 

July 

5 

5 

7 

Saturday 

Aug. 

1 

1 

Sept. 

4 

4 

Oct. 

6 

6 

Nov. 

2 

2 

Dec. 

4 

4 

where  K  =  5  n  +  r2,  Gregorian  ; 

#  =  0,     Julian. 

(Gregorian  K  =  20,  from  1900  to  1999.) 
Week-day  No.  =  r5  +  1. 


Note  16.     Gauss'  Rule  for  Easter  (p.  148). 

To  demonstrate  the  rule,  we  shall  consider  the  Julian  calendar 
first,  and  then  modify  our  results  to  accord  with  the  present 
Gregorian  calendar. 

The  lunar  month  of  chronology,  or  the  period  of  the  moon's 
orbital  revolution  around  the  earth,  is  approximately  29 \  days 
long.  In  making  the  ecclesiastical  calendar  it  was  therefore 
decided  to  have  lunar  months  of  29  and  30  days  occur  alternately 
as  a  general  rule.  But  for  a  reason  to  be  explained  in  a  moment, 
an  extra  lunar  month  of  30  days  is  inserted  at  the  end  of  every 
third  year  for  six  successive  periods  of  three  years  each,  or  eighteen 
years  in  all.  Then,  one  year  later,  at  the  end  of  the  nineteenth 
year,  an  additional  extra  lunar  month  of  29  days  is  further  in- 
serted in  the  calendar. 

2c  385 


ASTRONOMY 

The  lunar  calendar  for  nineteen  years  then  stands  as  follows  : 

3  years  (36  months)  alternating  29  and  30  days,  total        .     .     1062  days 

Extra  months  of  30  days       30  days 

The  above  repeated  five  times  more  (1092  x  5) 5460  days 

The  19th  year  of  12  months  alternating  29  and  30  days    .     .      354  days 

The  final  extra  month  of  29  days 29  days 

Total  6935  days 

The  above  calculation  takes  no  account  of  leap-years,  which 
occur  every  fourth  year  in  the  Julian  calendar.  To  get  these 
leap-years  into  the  lunar  calendar,  too,  the  ecclesiastical  chronol- 
ogists  adopted  the  simple  plan  of  putting  an  extra  day  into  the 
lunar  month  of  February,  whenever  it  is  put  into  the  civil  month  of 
February.  In  19  years  this  will  happen  five  times  when  any  one 
of  the  first  three  years  of  the  19  is  a  leap-year ;  but  if  the  fourth 
year  of  the  19  is  a  leap-year,  it  will  happen  four  times  only.  Thus, 
on  the  average : 

19  years  will  have    6935  +  5,  or  6940  days  three  times,  and 
19  years  will  have  6935  +  4,  or  6939  days  once. 

The  mean  of  these  figures  is  6939f  days ;  and  this  is  the  average 
number  of  days  in  19  lunar  years,  according  to  accepted  chronologic 
rules. 

Now  the  length  of  a  Julian  tropical  or  calendar  year  is  365J  days. 
Consequently,  19  Julian  years  will  contain  365 \  X  19,  or  6939f , 
days,  agreeing  exactly  with  the  lunar  figures  just  found.  This 
agreement  is  evidently  not  accidental,  but  is  the  result  of  the  above 
conventional  and  arbitrary  rules  for  the  extra  lunar  months. 

One  important  thing  follows  from  this  agreement :  if  we  write 
the  calendar  dates  of  full-moon  for  a  period  of  19  years,  these  calen- 
dar dates  will  then  be  repeated  in  the  next  and  in  every  subsequent 
period  of  19  years.  Now  it  so  happens  that  the  year  0,  or  the 
year  next  preceding  the  year  1  of  our  era,  was  the  first  year  of  a 
19-year  cycle.  Consequently,  the  year  1  was  the  second  of  the 
19-year  cycle,  the  year  2  the  third,  and  the  year  19  the  first  of  the 
next  cycle.  It  is  clear  that,  in  general,  if  we  divide  the  year 
number  y  by  19,  and  call  the  remainder  r6,  then  r6  +  1  will  be 
the  position  of  the  year  y  in  a  19-year  cycle. 

386 


APPENDIX 

The  next  step  is  to  find  for  any  year  the  date  of  the  Easter  full- 
moon,  which,  according  to  the  Nicene  council's  decree,  is  the  first 
to  fall  on  March  21  or  thereafter.  Let  us  call  the  date  of  this 
full-moon  March  21  -f  P,  and  suppose  dates  in  March  to  be  car- 
ried over  into  April,  so  that  April  1  will  be  called  March  32.  Now 
it  so  happens  that  in  the  year  preceding  the  year  1,  the  Easter 
full-moon,  Julian  calendar,  fell  on  March  36,  so  that  P  was  then  15. 
As  there  are  354  (12  X  29J)  days' in  a  lunar  twelve-month,  it  is 
clear  that  in  the  year  1  Easter  full-moon  must  have  occurred  11 
(which  is  365-354)  days  earlier.  And  in  each  succeeding  year  of 
the  19-year  period,  Easter  full-moon  must  have  occurred  either  11 
days  earlier  than  in  the  preceding  year,  or  19  (which  is  30-11) 
days  later.  Of  course  the"  occasions  when  it  occurs  19  days  later 
are  accounted  for  by  the  extra  30-day  months  inserted  every  three 
years.  The  following  table  exhibits  the  above  state  of  affairs : 


YEAR 

T6 

•    P 

0 

0 

15 

1 

1 

4 

11  days  earlier  than  year  0 

2 

2 

23 

19  days  later  than  year  1 

3 

3 

12 

11  days  earlier  than  year  2 

4 

4 

1 

11  days  earlier  than  year  3 

5 

5 

20 

19  days  later  than  year  4, 

etc.                                                                    etc. 

It  is  clear  that  we  shall  have,  in  general,  if  we  let  v  and  x  be 
two  unknown  whole  numbers : 

P  =  15  +  19  z  -  11  v, 
or: 

P  =  15  +  19  (x  -h  v)  -  30  v. 

It  is  further  clear  that  in  this  equation  : 

x  +  v  =  r6, 

because,  to  get  P  in  the  table  above,  we  have  always  added  19  r6and 
then  subtracted  the  largest  possible  value  of  30  v.  So  we  may 
write : 

P  =  19  r6  +  15  -  30  v. 

387 


ASTRONOMY 

From  this  it  appears  that  P  is  simply  the  remainder  occurring 
in  the  division  of  19  r&  +  15  by  30.  If  we  call  this  remainder  r7 
we  can  therefore  find  the  date  of  Easter  full-moon  in  the  Julian 
calendar  thus : 


Divide 

by 

and  call  the 
remainder 

Year  No.,  y, 
19  r6  +  15 

19 
30 

; 

And  the  date  of  Easter  full-moon, 
Julian  calendar,  is  March  21  +  r7. 

The  above  method  of  calculation  not  only  applies  to  the  first 
period  of  19  years,  but  is  entirely  general ;  because,  as  we  have  seen, 
subsequent  19-year  periods  simply  repeat  the  same  dates  of  full- 
moon. 

We  must  now  pass  to  the  Gregorian  calendar.  It  is  evident  that 
the  two  calendars  are  in  accord  at  the  beginning  of  the  era,  and 
do  not  diverge  until  the  year  100,  when  the  Gregorian  calendar 
omits  a  Julian  leap-year.  This  will  of  course  change  P  by  one  day, 
and  this  same  difference  of  one  day  will  continue  from  the  year  100 
to  the  year  199.  From  200  to  299  there  will  be  a  difference  of  two 
days,  etc. 

It  is  clear  that  we  can  allow  for  this  cause  of  difference  between 
the  two  calendars  by  varying  the  number  15  that  occurs  in  the 
quantity  19  r6  +  15.  Let  us  call  this  variable  number  M.  Then, 
in  both  calendars,  M  is  15  from  the  beginning  of  the  era  to  the  year 
99.  In  the  Gregorian  calendar  M  increases  by  1  each  century 
thereafter,  except  that  for  every  fourth  century  this  increase  is 
omitted  because  of  the  Gregorian  leap-year  exception.  Using 
our  former  notation,  in  which  c  is  the  century  number,  we  have  for 
the  Gregorian  calendar : 

M  =  15  +  c  -  i  (c  -  n). 

But  this  value  of  M ,  thus  corrected  for  the  Gregorian  leap-year, 
is  not  yet  quite  right.  A  further  last  correction  is  still  necessary 
on  account  of  a  slight  inaccuracy  in  the  lunar  period  of  19  years. 
A  lunar  month  is  not  exactly  29J  days  long;  its  true  length  is 
29.530586  days.  So  the  235  lunar  months  of  a  19-year  period 

388 


APPENDIX 

really  amount  to  235  X  29.530588  days ;  or  6939  days,  16  hours, 
31  minutes,  and  not  6939f  days,  as  already  obtained. 

The  error  of  1*  29W  amounts  to  a  day  in  308  years.  But  the 
framers  of  the  ecclesiastical  calendar  assumed  this  error  to  reach 
one  day  in  312  J  years,  or  8  days  in  2500  years.  So  they  directed 
that  a  correction  be  made,  such  that  M  be  diminished  by  1  seven 
times  successively  at  the  ends  of  300-year  periods,  and  an  eighth 
time  at  the  end  of  a  400-year  period,  or  8  times  in  2500  years. 
The  last  period  of  2500  years  terminated  at  the  end  of  the  year 
1799,  and  the  correction  was  then  5 ;  new  corrections  are  therefore 
required  in  2100,  2400,  2700,  3000,  3300,  3600,  3900,  all  at  in- 
tervals of  300  years.  But  the  next  following  correction  does  not 
come  until  4300  instead  of  4200,  on  account  of  the  eighth  period 
being  one  of  400  years.  This  condition  will  be  satisfied  for  all 
time  if  we  divide  8  c  +  13  by  25,  call  the  remainder  r8,  and  subtract 
from  M  the  correction  : 

8  c  +  13  -  r8 
.   25 

This  may  be  verified  readily  by  drawing  up  a  table  of  this  cor- 
rection, when  it  will  be  found  to  have  the  value  5  for  y  =  1799, 
and  to  increase  thereafter  forever  in  the  proper  way.  We  have, 
then,  finally,  for  the  Gregorian  calendar: 

M  =  15  +  c  -  i  (c  -  n)  -  A-  (8  c  +  13  -  rg)  ; 
and  wnen  M  is  greater  than  30,  we  may  subtract  from  it,  if  we 
choose,  the  largest  possible  exact  multiple  of  30.  And  in  the 
Gregorian  calendar  the  date  of  Easter  full-moon  is  now  March 
21  +  r7,  where  r7  is  the  remainder  resulting  from  the  division  by 
30  of  the  number  19  r6  +  M. 

Having  thus  found  a  method  of  calculating  the  Gregorian  date, 
March  21  +  r7,  of  the  Easter  full-moon,  we  must  now  find  the 
date  of  the  Sunday  next  following,  which  will  be  Easter  Sunday. 
We  need  therefore  only  calculate  the  week-day  of  the  date  March 
21  +  r7,  to  know  the  date  of  Easter.  Referring  to  our  former 
civil  calendar  formulas,  we  shall  find  the  remainder  r5  for  the  Easter 
full-moon  date,  March  21  +  r7,  which  remainder  we  shall  call 
TQ  for  this  special  case,  if  we  divide  by  7  the  quantity  : 

5  r3  +  3  r4  +  K  +  2  +  r7. 
389 


ASTRONOMY 


Now  if  r9  comes  out  0,  the  Easter  full-moon  comes  on  Sunday, 
and  Easter  is  7  days  later,  according  to  the  Nicene  decree.  If  r9  is 
1,  the  full-moon  day  is  Monday,  and  Easter  is  6  days  later.  In 
general,  we  obtain  the  date  of  Easter  Sunday  by  adding  to 
March  21  +  r^  the  number : 

7  -r9. 

Collecting  all  our  formulas,  we  can  now  find  the  date  of  Easter 
Sunday  as  follows ;  and  thus  the  rule  given  on  p.  148  is  demon- 
strated. 


Divide 

by 

and  call 
remainder 

Century  No.,  c 

4 

r\ 

K  =  5  ri  -f-  r2,  Gregorian  calendar. 

Century  No.,  c 

7 

r-i 

K  =  0,  Julian  calendar. 

Year  No.,  y 

4 

r* 

M  =  15  +  c  -  1  (c  -  n) 

-•&  (8  c  +  13  -  r8),  Greg,  calen- 

Year No.,  y 

7 

r* 

dar.     M  =  15,  Julian  calendar. 

8c  +  13 

25 

r$ 

Easter    Sunday    is    then    March 

Year  No.,  y 

19 

r& 

28  +  r7  -  r9, 

19  r6  +  M 

30 

TI 

or  April  r7  —  r9  —  3. 

5  r3  +  3  n  +  K  \ 

+   O      1 

7 

7-9 

The  following  are  values  of  K  and 

2  +  r-i  } 

M  for  the  Gregorian  calendar  : 

1800-1899,   K  =  14,   M  =  23, 

1900-1999,   K  =  20,   M  =  24. 

As  an  example,  let  us  calculate  the  date  of  Easter  Sunday  for 
1913.  We  have : 

r3  =  1,  r4  =  2,  K  =  20,  M  =  24,  r6  =  13,  r7  =  1,  r9  =  6; 

Easter  Sunday  is  on  March  28  +  1  -  6  =  March  23. 

We  must  now  explain  the  two  exceptions  that  occur  in  the 
Gregorian  calendar  only  (p.  149).  The  first  of  these  happens  when 
r-i  =  29.  The  formulas '  have  been  deduced  on  the  supposition 
that  the  March  and  April  full-moons  occur  at  an  interval  of  30 
days.  But  that  interval  may  be  29  days  only.  The  framers  of 
the  calendar  have  assumed,  rather  arbitrarily,  that  if  there  is  a 
full-moon  on  March  19,  or  earlier  in  March,  the  April  full-moon 

390 


APPENDIX 

will  occur  30  days  later.  But  if  the  March  full-moon  is  on  the 
20th,  or  later,  the  April  full-moon  will  happen  29  days  later. 
Thus  the  ecclesiastical  April  full-moon  will  happen  on  the  same 
day,  no  matter  whether  the  March  full-moon  comes  on  the  19th 
or  20th. 

As  this  cannot  occur  in  reality,  the  framers  of  the  calendar  have 
directed  that  when  the  March  full-moon  happens  on  the  20th, 
which  occurs  whenever  r7  =  29,  then  r7  shall  be  diminished  arbi- 
trarily by  1.  That  is,  we  must  use  28  instead  of  29,  or  move  the 
April  moon  back  one  day.  But  the  diminution  of  r7  by  1  will 
ordinarily  also  diminish  r9  by  1.  Consequently,  r7  —  rg  will 
remain  unchanged,  and  so  will  the  date  of  Easter  Sunday,  which 
depends  on  r7  —  r9.  Only  when  r7.  =  29  and  r9  =  0  will  the 
change  of  r7  from  29  to  28  have  any  effect.  For  when  r9  =  0, 
a  diminution  by  1  will  change  it  into  6,  and  r7  —  r9  will  be  diminished 
by  7,  making  Easter  exactly  one  week  earlier.  But  when  r7  =  29 
and  r9  =  0,  the  rule  always  makes  Easter  come  on  April  26. 
Therefore  the  exception  is  as  stated :  whenever  Easter  comes  by 
the  rule  on  April  26,  use  April  19  instead.  There  will  be  an  example 
of  this  in  1981. 

Unfortunately,  the  above  exceptional  case  introduces  another 
complication.  The  change  of  r7  from  29  to  28  does  not  make  it 
impossible  for  the  value  r7  =  28  to  occur  again  under  the  general 
rule,  and  during  the  same  19-year  period.  This  might  make  two 
full-moons  occur  on  the  same  date  twice  in  a  single  19-year 
period,  which  is,  in  fact,  impossible.  To  avoid  this,  the  framers  of 
the  calendar  have  ruled,  again  arbitrarily,  that  there  shall  be  a 
second  exception.  Under  this  exception,  28  is  changed  to  27, 
whenever,  in  the  same  19-year  period,  the  first  exception  occurs. 

We  must  therefore  investigate  when  the  first  exception  can 
occur.  In  determining  r7  we  performed  a  division  by  30.  Let  us 
indicate  the  quotient  of  this  division  by  v.  Then  we  have,  if  r7 
is  29,  according  to  the  first  exception : 

19  r6  +  M  =  30  v  +  29. 

Now  multiply  this  equation  by  11,  and  add  11  to  each  member. 
This  gives : 

209  r6  +  11  M  -f  11  =  330*;  +  319  +  11  =  330  v  +  330. 

391 


ASTRONOMY 

The  right  hand  member  is  now  divisible  exactly  by  30 ;  there- 
fore the  left-hand  member  is  also  so  divisible.  But  the  division  of 
209  r6  by  30  will  leave  a  remainder  of  29  r&.  To  make  this  dis- 
appear, the  remainder  in  the  division  of  1 1  M  +  1 1  by  30  must  be 
r6.  But  r6  is  always  less  than  19  by  its  definition.  Therefore  the 
first  exception  will  occur  whenever,  in  the  division  of  11  M  +  11 
by  30,  the  remainder  is  less  than  19. 

But  again,  as  in  the  case  of  the  first  exception,  the  change  of 
TI  from  28  to  27  will  make  no  difference  in  the  date  of  Easter, 
unless  r9  =  0.  When  r7  =  28  and  r9  =  0,  Easter,  according  to  the 
rule,  comes  on  April  25.  The  change  of  r7  moves  this  date  to 
April  18.  Therefore  the  second  exception  reads :  whenever,  in 
the  division  of  11  M  +  11  by  30,  the  remainder  is  less  than  19,  and 
if  r-i  =  28  and  r9  =  0,  Easter  Sunday  is  on  April  18,  instead  of 
April  25,  as  given  by  the  rule.  An  example  of  this  will  occur  in 
1954. 

To  complete  this  subject  it  is  necessary  to  remark  that  r7  can 
never  be  29,  28,  and  27  within  a  single  period  of  19  years.  There- 
fore no  further  exception  is  necessary  on  account  of  the  possibility 
that  the  above  two  exceptions  might,  by  acting  together,  produce 
two  cases  of  r7  =  27  in  a  single  19-year  period. 

Note  17.     The  Sextant  (p.  154). 

To  prove  the  fundamental  principles  of  the  sextant,  that  the 
angle  between  the  mirrors  is  half  the  altitude  of  the  sun,  imagine 
the  plane  of  the  paper  to  be  the  plane  of  the  circle  of  the  sextant. 
Then,  in  Fig.  113,  the  plane  of  the  circle  is  supposed  to  be  held 
vertically,  in  such  a  way  that  it  will  pass  through  the  sun  at  S. 
The  navigator  sees  the  horizon  with  the  upper  part  of  the  telescope 
through  the  unsilvered  part  of  the  mirror  m ;  and  he  sees  the  sun 
along  the  line  TmMS  after  reflection  from  both  mirrors.  The 
angle  MTm  is  the  altitude  of  the  sun  above  the  horizon;  the 
angle  at  P  is  the  angle  between  the  mirrors.  It  is  necessary  to 
prove  that : 

Angle  MTm  =  2  angle  P. 

The  lines  MP'  and  mP'  are  drawn  perpendicular  to  the  mirrors 
M  and  m.  Then,  according  to  the  optical  principles  governing 

392 


APPENDIX 


the  Horizon    \ 


FIG.  113.    Theory  of  Sextant. 

the  reflection  of  light  from  plane  mirrors,  the  two  angles  a  are 
equal  and  so  are  the  two  angles  b.  Furthermore,  the  angle  SMm, 
or  2  a,  is  an  exterior  angle  to  the  triangle  mMT.  Consequently  : 

Angle  2  a  =  angle  26  +  angle  M  Tm, 
or : 

Angle  MTm  =  2  (angle  a  —  angle  b). 

Similarly,  from  the  triangle  mMP' : 

Angle  P'  =  angle  a  —  angle  b. 

But  arigle  P'  =  angle  P,  because  their  sides  are  perpendicular, 
each  to  each.  Therefore, 

Angle  P  =  angle  a  —  angle  6. 
And  it  follows  that : 

Angle  MTm  =  2  angle  P. 

Q.  E.  D. 

393 


ASTRONOMY 

Note  18.     Longitude  Determination  (p.  158). 

A  reference  to  Fig.  106  (p.  364)  will  show  how  the  longitude  may 
be  computed  from  the  sun's  altitude,  measured  with  the  sextant. 
In  the  figure,  if  Sf  is  the  sun,  the  arc  AS'  is  the  measured  altitude. 
If  this  be  subtracted  from  90°,  we  have  the  arc  ZS',  or  angular 
zenith  distance  of  the  sun.  The  arc  S'D  is  the  sun's  declination, 
and  may  be  ascertained  for  the  date  of  the  observation  from  the 
nautical  almanac.  Subtracting  this  declination  from  90°  makes 
known  the  arc  PS',  or  the  angular  polar  distance  of  the  sun. 

The  ship's  latitude  is  also  supposed  to  be  known;  without  it, 
the  longitude  cannot  be  computed.  But  the  ship's  latitude 
always  is  known,  because  the  navigator  will  have  determined  it 
at  noon,  and  can  easily  allow  for  any  slight  change  in  the  ship's 
latitude  since  the  last  noon  observation,  since  he  knows  the  com- 
pass course  he  is  steering,  and  the  approximate  speed  of  the  ship. 

But  the  latitude  is  the  arc  PN  in  the  figure,  or  the  altitude  of 
the  celestial  pole  above  the  horizon.  This  latitude  being  subtracted 
from  90°,  gives  the  arc  ZP,  or  the  angular  distance  from  the  ce- 
lestial pole  to  the  zenith.  Thus  these  three  subtractions  from  90° 
make  known  the  three  sides  of  the  spherical  triangle  ZPS'. 

It  is  a  principle  of  trigonometry  that  any  spherical  triangle  can 
be  solved  completely,  and  all  its  parts  made  known,  if  we  know 
its  three  sides.  Thus  we  find  the  spherical  angle  S'PZ,  of  which 
the  vertex  is  at  the  pole,  and  which  is  measured  by  the  arc  DM  on 
the  celestial  equator.  But  DM  is  by  definition  the  hour-angle  of 
the  sun  S' ;  and  the  sun's  hour-angle  is  the  local  apparent  solar 
time.  This  need  merely  be  corrected  by  applying  the  equation  of 
time  (p.  134)  to  obtain  the  local  mean  solar  time  of  the  ship,  ready 
for  comparison  with  the  Greenwich  time  taken  from  the  face  of 
the  chronometer  by  an  assistant  at  the  instant  when  the  sun  was 
observed  for  longitude  by  the  navigator. 

Note  19.     Moon's  Distance  (p.  169). 

Figure  114  shows  how  the  moon's  distance  is  determined.  We 
shall  assume,  as  a  sufficiently  close  first  approximation,  and  to  make 
the  problem  easy  to  understand,  that  the  two  observatories  are 
situated  on  the  same  meridian  of  terrestrial  longitude,  but  very 
widely  separated  in  latitude.  One  should  be  in  the  northern 

394 


APPENDIX 

hemisphere;  the  other  in  the  southern.  The  observatories  of 
Greenwich,  England,  and  the  Cape  of  Good  Hope,  for  instance, 
satisfy  these  conditions  quite  closely. 

In  Fig.  114,  then,  0  and  0'  are  the  two  observatories,  the  circle 
representing  the  earth.  The  arc  00'  is  known,  for  it  is  simply 
the  latitude  difference  of  the 
two  observatories.  The  angle 
OCO'  is  equal  to  the  arc  00' ; 
and  [the  lines  CO  and  CO'  are 
each  known  radii  of  the  earth. 
Therefore,  by  simple  trigo- 
nometry, we  can  solve  the  tri- 
angle OCO',  and  gain  a  knowl- 
edge of  the  distance  00',  which 
is  to  be  our  base-line,  and  of 
the  two  angles  COO'  and  CO'O. 

We  next  measure  at  both 
observatories  simultaneously, 
with  suitable  astronomic  in- 
struments, the  exact  lunar  alti- 
tude, or  angular  elevation  of 
the  moon  above  the  horizon,  at 
the  instant  when  the  diurnal 
rotation  has  brought  the  moon  to  the  celestial  meridian.  These 
simultaneous  observations  will  be  possible,  because  the  moon  will 
reach  the  meridian  of  both  places  at  the  same  instant,  since  we 
have  imagined  our  two  observatories  lying  on  the  same  meridian 
of  terrestrial  longitude,  and  therefore  having  the  same  celestial 
meridian  over  them  in  the  sky. 

Having  measured  the  moon's  altitude  above  the  horizon,  we  can 
at  once  find  its  angular  distance  from  the  zenith.  For  the  latter 
point  is  always  90°  distant  from  the  horizon ;  so  that  we  obtain 
the  angular  zenith  distance  of  the  moon  by  simply  subtracting 
its  measured  altitude  from  90°. 

Those  two  angular  zenith  distances,  thus  known  from  the 
measured  altitudes,  are  the  angles  MOZ  and  MO'Z'  in  Fig.  114. 
Next  we  subtract  these  angles  from  180°,  giving  us  the  angles  MOC 

395 


FIG.  114.     Moon's  Distance. 


ASTRONOMY 

and  MO'C.  From  these  we  again  subtract  the  angles  COO'  and 
CO'O,  found  above,  thus  obtaining  values  of  MOO'  and  MO'O. 
These  now  make  possible  a  trigonometric  solution  of  the  triangle 
MOO',  of  which  we  now  know  the  base  00'  and  the  two  adjoining 
angles.  Thus  we  get  OM  and  O'M  in  miles.  After  that  we  can 
solve  the  two  triangles  COM  and  CO'M,  since  we  know  the  length 
of  the  two  sides  CO  and  OM,  as  well  as  the  included  angle  COM ; 
and  in  the  other  triangle  we  know  CO'  and  O'M  as  well  as  the  in- 
cluded angle  CO'M.  A  solution  of  either  triangle  gives  us  CM, 
the  distance  from  the  center  of  the  earth  to  the  moon. 

It  is  scarcely  necessary  to  add  that  the  ideal  condition  here 
assumed  as  to  location  of  observatories  does  not  exist  in  fact.  But 
a  slight  divergence  from  this  condition  in  no  way  impairs  the  prin- 
ciple of  the  method;  it  merely  adds  a  certain  additional  com- 
plexity to  the  trigonometrical  calculations. 

Note  20.     Lunar  Parallax  (p.  169). 

Figure  42  shows  that  the  moon's  parallax  and  distance  are  con- 
nected by  a  very  simple  trigonometric  formula : 

°f 


sin  parallax  =  -^—-  = 

fLM     distance  of  moon 

This  formula  shows  that  we  can  calculate  the  parallax  if  we  know 
the  distance,  or  the  distance  if  we  know  the  parallax.  The  two 
are  closely  related;  astronomers  frequently  speak  of  measuring 
the  parallax  of  the  moon  or  other  heavenly  body,  when  they 
merely  mean  a  measurement  of  its  distance. 

Note  21.     The  Moon's  Mass  (p.  175). 

Figure  115  is  intended  to  make  this  matter  clear.  S  is  the  sun ; 
the  circle  is  the  annual  terrestrial  orbit.  When  the  center  of 
gravity  is  at  Ci,  the  earth  at  EI,  and  the  moon  at  MI,  the  sun  will 
appear  from  the  earth  projected  in  the  direction  Si.  This  is  exactly 
the  same  as  would  be  the  case  if  there  were  no  moon,  for  then  the 
earth  would  itself  be  at  C\.  •  But  when  the  center  of  gravity  is  at 
C2,  the  earth  will  be  at  E2 ;  and  the  sun  will  be  seen  projected  in 
the  direction  $2',  instead  of  $2,  which  is  its  direction  as  seen  from 
C2,  and  which  would  be  its  direction  from  the  earth  if  there  were 
no  moon. 

396 


APPENDIX 


Thus  the  sun  will  be  seen  a  certain  angular  distance  in  advance 
of  its  proper  position ;  and  a  half-month  later  it  will  be  similarly 
retarded.  The  total  range  is  12",  so  that  the  angle  S2SS2  is  6". 
Therefore,  in  the  triangle  C2SE2,  we  know  the  angle  C2SE2  to  be 
6";  and  we  know 
the  two  sides  C2S 
and  E2S,  the  dis- 
tance from  the 
earth  to  the  sun, 
which  can  be  meas- 
ured.  Solving 
the  triangle,  we 
find  the  side  C2E2 
to  be  about  2880 
miles.  We  then 
form  the  propor- 
tion : 

E2C2  :  M2C2  = 
moon's  mass : 
Earth's  mass ; 

from  which  we  can 
compute  the  lunar  mass,  since  the  other  quantities  in  the  propor- 
tion are  now  all  known. 

Note  22.     Concavity  of  Moon's  True  Orbit  with  Respect  to  the 

Sun   (p.  181). 

We  can  test  this  question  by  means  of  Fig.  116.  It  is  evident 
from  a  mere  glance  at  Fig.  46  (p.  181)  that  there  is  no  doubt  as  to 
the  concavity  of  the  moon's  orbit  toward  the  sun  at  the  time  of 
full-moon,  shown  at  Ms.  Difficulty  arises  only  in  the  case  of  the 
new-moon  phase,  shown  at  MI  and  M5.  Therefore,  in  Fig.  116, 
we  shall  examine  especially  the  new-moon  phase.  Let  Eit  MI,  and 
S  be  positions  of  the  earth,  moon,  and  sun  at  the  time  of  new-moon. 
Let  EiE2  be  a  portion  of  the  earth's  orbit  around  the  sun ;  and  let 
the  small  circles  represent  the  lunar  orbit  around  the  earth.  While 
the  earth  moves  from  E\  to  E2,  we  may  suppose  the  moon  to  move 
around  the  earth  from  C  to  M2.  In  other  words,  if  the  moon  did 

397 


FIG.  115.     Mass  of  the  Moon. 


ASTRONOMY 


not  revolve  around  the  earth,  it  would  be  at  C  when  the  earth 
reached  E2.  Designate  the  angle  EiSE2  by  the  letter  0,  and  let 
re  and  rm  represent  radii  of  the  earth's  orbit  around  the  sun  and 
the  moon's  orbit  around  the  earth.  Finally,  let  EiT  be  a  tangent 

to  the  earth's  orbit  at 
El]  draw  E2P  perpen- 
dicular to  #1$;  E2B 
parallel  to  EiS;  and 
M2A  parallel  to  E^T. 
If  we  let  0  be  a  small 
angle,  MiM2  will  be  a 
small  part  of  the  moon's 
path  near  new-moon : 
it  will  evidently  be  con- 
cave towards  the  sun  if 
M2  is  farther  from  the 
tangent  EiT  than  is  M\. 
While  the  moon  was 
moving  from  MI  to  M2 
the  entire  lunar  orbit 
fell  away  from  the  tan- 
gent the  distance  E\P ; 
but  the  moon  rose  toward  the  tangent  a  distance  nearly  equal  to 
AB.  Therefore  the  moon  recedes  from  the  tangent  a  total  dis- 
tance of  EiP  —  AB.  Now  we  have,  evidently : 


FIG.  116.     Moon's  True  Orbit. 


But: 

Also : 


EiP  =  re  -  re  cos  6, 
AB  =  rm  —  rm  cos  M2E2A. 
M2E2A  =  M2E2C  +  CE2A. 


(1) 

(2) 


M2E2C  =  13  6,  because  the  moon's  angular  motion  in  its  orbit 
is  about  13  times  as  fast  as  the  earth's  (p.  161) ;  and  : 

CE2A  =  6,  because  AB  is  parallel  to  E& 
It  follows  that:  M^A  =  140. 

and,  from  equation  (2) : 

AB  =  rm- rwcosl40.  (3) 


PHILOSOPHISE 

N  A  T  U  R  A  L  I  S 

PR  INC  I  PI  A 

MATHEMATICA 


Autore  J  S  N  E  ffTO  N,  Inn.  Coll.  Cantab.  Sac.    Mathefcos 
Profeflbre  LncaftanO)    Be  Socictatis  Regalis  Sodali. 


IMPRIMATUR- 

S.    P  E   P   Y  S,     Reg.  Sac.   P  R  JE :  S  E 
y*iii   5.  1686. 


L  0  N  D  I  N  /, 

fuflii  Soeietatif  Regime  ac  Typis  Jofefhi  Streater.   Profiat  apud 
plures  Bibliopolas.    Anno  MDCLXXXVIL 


PLATE  32.     Title-page  of  Newton's  Principia. 


APPENDIX 


A   simple   calculation,  using  the  value  0  =  1( 
r2  =  93,000,000,  gives : 

EiP  =  16,000  miles, 
AB  =  7130  miles. 


rm  =  240,000, 


It  follows  that  the  moon  recedes  from  the  tangent  about  8870 
miles  in  one  day,  while  the  earth  is  moving  about  1°  in  its  orbit 
around  the  sun.  This  proves  that  the  moon's  true  orbit  is  con- 
cave towards  the  sun,  even  at  the  time  of  new-moon. 


Note  23.     Law  of  Areas  (p.  186). 
Figure  117  shows  how  the  point 


P3'  is  found.      Draw 


parallel  and  equal  to  P2P2'.     Then  the  actual  motion  of  the  planet 

in  the  second  second  will  take  place  along  the 

diagonal  P2P3'  of  the  parallelogram  P2P3P3'P2'. 

This  theorem  of  the  " parallelogram  of  forces"  is 

demonstrated  in  works  on  elementary  physics : l 

perhaps  the  easiest  way 'to  understand  it  is  to 

notice  that  P3'  is  point  to  which  P2  must  go,  if 

it  actually  completes  separately  the  two  motions 

P2P2',  and  P2'P3'  equal  and  parallel  to  P2P3. 


FIG.  117.     Law  of 
Areas. 


Note  24.     Law  of  Areas  (p.  186). 2 

We  have  still  to  prove  the  triangles  SPiP2 
and  *SP2P3'  equal  in  area.  Referring  again  to 
Fig.  117,  we  see  that  the  triangles  $P2P3'  and  $P2P3  are  equal,  since 
they  have  the  same  base  £P2,  and  their  altitudes  are  equal  because 
their  vertices  P3  and  P/  lie  on  the  line  P3P3',  which  is  parallel 
to  P2*S.  And  we  have  already  found  the  triangle  $P2P3  equal  to 
SPiP2.  Therefore  the  triangle  SP2P3'  is  also  equal  to  £PiP2. 

1  Figure  117  may  be  found  in  the  first  edition  of  Newton's  immortal  Prin- 
cipia,  of  which  the  title-page  is  reproduced  as  Plate  32.     The  president  of 
the  Royal  Society,  whose  name  appears  on  the  title-page  as  having  author- 
ized the  printing,  is  the  famous  diarist.      On  p.  13  of  the  Principia  ap- 
pears Corol.  I :   "  Corpus  viribus  conjunctis  diagonalem  parallelogrammi 
eodem  tempore  describere,  quo  latera  separatis." 

2  On  p.  37  of  the  same  work  of  Newton  appears  Prop.  I,  Theorem  I : 
"Areas  quas  corpora  in  gyros  acta  radiis  ad  immobile  centrum  virium 
ductis  describunt  .  .  .  esse  temporibus  proportionates." 

399 


ASTRONOMY 


Note  25.     Harmonic  Law  (p.  188). 

It  would  carry  us  too  far  afield  in  mathematical  astronomy  to 
give  here  the  demonstrations  by  which  Kepler's  three  laws  may  be 
derived  from  Newton's  single  law ;  but  there  is  little  difficulty  in 
considering  by  elementary  methods  the  special  case  of  a  circular 
planetary  orbit.  The  circle  is,  in  fact,  a  close  approximation  to 
the  actual  planetary  orbits  in  the  solar  system  :  none  of  these  orbits 

are  very  much  flattened  from  the 
circular  form. 

We  must  first  investigate  the 
nature  of  the  solar  attractive  force. 
In  the  case  of  a  circular  orbit  this 
force  is  necessarily  constant  under 
Newton's  law,  because  the  planet 
is  always  at  the  same  distance 
from  the  sun.  Now  consider  the 
accompanying  Fig.  118.  Let  PP' 
be  a  very  short  arc  of  a  circle, 
whose  center  is  at  S.  Draw  the 
diameter  PD  and  the  chord  P'D ; 
and  let  fall  the  perpendicular  P'C 
upon  PD.  Draw  the  chord  PP',  the  tangent  PP";  and  let  fall 
the  perpendicular  P'P"  upon  PP"  from  Pf.  Then,  from  the 
similar  right-angled  triangles  PP'C  and  PP'D,  we  have : 


FIG.  118.    Solar  Attraction. 


or: 


PP'  :  PC  =  PD  :  PP', 
PP72 


PC  = 


PD 


Now  let  our  circle  be  a  planetary  orbit,  with  the  planet  at  P, 
the  sun  at  S ;  and  suppose  that  in  one  second  of  time  the  planet 
would  move  along  the  orbit  to  P' .  We  may  consider  this  very 
short  arc  PP'  coincident  with  its  chord  PP'. 

From  the  principle  of  the  parallelogram  of  forces  (p.  399),  the 
actual  motion  PP'  may  be  regarded  as  the  resultant  of  two 
motions :  PP",  which  would  be  the  planet's  actual  motion  from 
P  in  a  second  if  the  solar  attraction  were  to  cease  suddenly ;  and 

400 


APPENDIX 

PC,  which  would  be  the  planet's  actual  motion  in  a  second  if 
attraction  toward  the  sun  operated  alone. 

Now  PP'  is  the  planet's  velocity  in  its  orbit  per  second,  which  we 
shall  call  V  ;  and  PD  is  twice  the  radius  of  the  orbit,  which  latter  we 
shall  call  r.  Let  us  also  designate  the  distance  PC  by  the  letter  x, 
and  consider  all  distances  to  be  measured  in  miles.  Then,  from 
the  geometry  of  the  figure,  as  we  have  just  seen  : 


But  as  we  have  said,  PC  or  x  is  the  distance  the  planet  would 
move  or  "fall"  toward  the  sun  in  a  second,  if  the  solar  attraction 
acted  alone,  without  any  additional  orbital  or  tangential  impulse 
derived,  perhaps,  from  the  original  catastrophe  by  which  the 
planet  was  brought  into  existence.  The  question  now  is  :  How 
great  must  be  the  solar  attractive  force  to  cause  a  planet  to  fall 
from  a  position  of  rest  at  P  through  the  distance  PC  or  x  in 
a  second  ? 

This  raises  the  question  of  how  forces  are  measured.  What  is  a 
suitable  unit  of  force?  Now  the  solar  attraction  is  not  applied 
suddenly  as  a  single  impulse  ;  it  is  applied  continuously.  Conse- 
quently, the  planet  would  fall  the  short  distance  x  toward  the  sun 
with  a  uniformly  increasing  velocity,  faster  and  faster,  but  begin- 
ning with  zero  velocity  at  P.  Its  average  velocity  would  be 
attained  halfway  between  P  and  C.  But  the  actual  distance  it 
would  move  in  a  second  is  of  course  the  same  as  if  it  traveled 
constantly  with  its  average  velocity.  And  as  it  would  fall  a 
distance  x  miles  in  a  second,  its  average  velocity  must  be  x  miles 
per  second.  Therefore  it  would  be  moving  with  the  velocity 
x  miles  per  second  when  halfway  between  P  and  C  ;  and  upon 
reaching  C  its  velocity  would  have  increased  to  2  x  miles  per  second. 
But  in  astronomy,  as  in  mechanics,  our  units  are  so  chosen  that 
force  is  always  measured  by  the  quantity  of  velocity  accumulated 
in  a  second,  multiplied  by  the  quantity  of  mass  in  the  moving  body. 
The  velocity  thus  accumulated  in  a  second  is  called  ''acceleration"  ; 
and  as  the  velocity  of  the  falling  planet  increased  from  zero  to  2  x 
miles  per  second,  the  acceleration  produced  by  the  solar  attractive 
2D  401 


ASTRONOMY 

force  must  be  represented  by  the  number  2x.     Calling  this  ac- 
celeration /,  we  thus  have  : 

/=2z; 

and  this,  combined  with  equation  (1),  gives  : 

172 

/  -  V--  (2) 

Now  the  whole  circumference  of  the  circular  orbit  is  2  irr  ;  and 
the  planetary  orbital  velocity  V  is  of  course  equal  to  the  circum- 
ference divided  by  the  period  of  orbital  revolution.  It  follows 
that  if  we  call  this  period  t,  expressed  in  seconds  of  time,  we  shall 
have: 


and,  therefore,  from  equation  (2)  : 

PI   111  ijjj:.    f=^r-         \  (3) 

If  we  now  apply  equation  (3)  to  two  separate  planets,  indicating 
by  subscript  numbers  quantities  belonging  to  the  first  and  second 
of  the  two  planets,  we  shall  have  : 


or  : 


But  we  know  from  Newton's  law  that  the  attractive  forces 
exerted  by  the  sun  on  two  different  planets,  if  of  equal  mass,  will 
be  inversely  proportional  to  the  squares  of  the  distances  separating 
those  planets  from  the  sun.  This  may  be  written  thus  : 

/i:A-tf:ri*; 
or: 

'  s-j  <5> 

Equating  the  right-hand  members  of  equations  (4)  and  (5)  gives  : 

*i2  :  tf  = 
402 


APPENDIX 

This  is  the  third  (or  harmonic)  law  of  Kepler,  which  is  therefore 
thus  demonstrated  as  a  consequence  of  Newton's  law  in  the  case 
of  circular  orbits.  A  similar  proof  is  possible,  by  the  aid  of  the 
higher  mathematics,  without  this  assumption  as  to  the  form  of 
the  orbit;  but  a  small  correction  is  always  required,  because  we 
have  taken  the  planets  to  have  equal  masses. 

Still  retaining  our  circular  orbit  formulas  as  a  sufficient  first 
approximation,  we  are  now  in  a  position  to  understand  Newton's 
famous  test  as  to  whether  the  force  of  gravitation  observable  on 
the  earth  also  extends  outward  as  far  as  the  moon.  We  shall 
present  this  test  here  in  a  somewhat  modernized  form,  based  on 
the  formulas  just  obtained.  Resuming  our  equation  (3),  we  have 
the  acceleration  exerted  by  the  earth  upon  the  moon  : 


in  which  r  is  now  the  distance  from  the  earth  to  the  moon,  and  t  the 
moon's  sidereal  period  (p.  161).  This  equation  is  correct,  if  the 
Newtonian  law  of  gravitation  extends  to  the  moon,  and  not  other- 
wise. 

Newton's  test  now  consists  in  comparing  the  value  of  /  calculated 
by  means  of  equation  (6)  with  its  value  easily  obtained  by  another 
method.  It  was  known  from  laboratory  experiments,  even  in 
the  time  of  Newton,  that  the  earth  attracts  an  object  situated  on 
its  surface  with  a  force  which  is  called  the  "  force  of  gravity,"  and 
which  produces  an  acceleration  designated  by  the  letter  g  in  physics. 
It  is  also  known  that  the  earth's  attraction  upon  any  object 
exterior  to  it  acts  as  if  the  entire  mass  of  the  earth  were  concen- 
trated at  its  center.1 

Now  the  distance  from  the  earth's  center  to  an  object  on  its 
surface  is  equal  to  the  earth's  radius,  and  may  be  designated  by 
R  ;  while  the  distance  from  the  earth's  center  to  the  moon  is  r. 
It  follows  that  if  the  earth's  attraction  varies  inversely  as  the 
square  of  its  distance  from  the  object  attracted,  as  postulated  by 
Newton,  we  may  write  the  following  simple  proportion  involving 

1  This  was  demonstrated  by  Newton. 
403 


ASTRONOMY 

/,  due  to  the  earth's  attraction  upon  the  moon,  and  g.  due  to  the 
earth's  attraction  upon  surface  objects  : 

f:g=7*:  R?' 

from  which  we  have  at  once  : 


The  values  of  /  in  equations  (6)  and  (7)  must  be  equal,  if  both  g 
and  /  result  from  the  same  identical  law  of  Newtonian  gravitation. 
Equating  these  quantities  gives  : 


or  : 

(8) 


In  this  equation,  r  is  the  moon's  distance  from  the  earth,  which 
we  here  suppose  expressed  in  feet;  and  t  is  the  moon's  sidereal 
period,  in  seconds  of  time.  Let  us  then  calculate  g,  and  ascertain 
whether  it  agrees  with  its  known  value  derived  by  physicists  from 
laboratory  experiments.  The  moon's  sidereal  period  is  27d  7h  43m 
11.5s,  or  2360591.5  seconds.  The  moon's  distance,  r,  is  238,840 
miles,  or  1,261,075,200  feet.  The  earth's  radius  is  3858.8  miles, 
or  20,902,464  feet.  The  value  of  TT  is  3.1416.  Making  the  calcu- 
lation by  means  of  logarithms,  the  above  data  give,  by  the  aid 
of  equation  (8)  : 

g  =  32.5, 

which  is  in  very  close  accord  with  the  value  of  g  found  directly  by 
experiment  in  the  physical  laboratory.  It  is  a  most  astounding 
thing  that  a  series  of  quantities  can  thus  be  brought  together, 
as  it  were,  from  various  parts  of  the  solar  system  :  the  moon's 
distance  determined  by  astronomic  observations  at  Greenwich 
and  the  Cape  of  Good  Hope  (p.  395)  ;  the  earth's  radius  by  triangu- 
lation  measures  (p.  97)  ;  the  moon's  period  by  noting  the  interval 
between  distant  full-moons  (p.  162),  —  it  is  astounding  that  these 
heterogeneous  quantities,  thus  determined  by  direct  observation, 
can  be  combined  by  a  simple  formula  based  on  that  wonderful  law 

404 


APPENDIX 

of  Newton,  and  made  to  produce  the  identical  value  for  g  which 
we  obtain  by  terrestrial  laboratory  observations,  quite  without 
using  astronomic  material.  There  could  not  be  a  more  striking 
proof  of  the  unity  of  science  ;  nor  can  any  doubt  remain  that  the 
same  force  of  gravity  which  controls  experiments  on  the  earth, 
also  controls  the  moon's  orbital  motion. 

Note  26.     Planet's  Mass  (p.  205). 

Let  us  suppose  once  more  that  orbits  are  all  circular.  Consider- 
ing the  satellite  orbit,  we  found,  when  discussing  Newton's  test 
of  the  law  of  gravitation  by  means  of  the  moon,  that  the  accelera- 
tion due  to  the  attractive  force  toward  the  center  of  the  orbit  may 
be  represented  by  the  equation  (p.  403)  : 


where  r  is  now  the  radius  of  the  satellite's  orbit  in  miles,  and  t  its 
period  of  revolution.  In  this  equation,  /  is  due  to  a  continuously 
acting  attractive  force  toward  the  planet  situated  in  the  center 
of  the  orbit,  supposed  circular. 

It  is  easy  to  obtain  another  expression  for  this  force.  We  have 
at  once,  from  Newton's  law  of  gravitation  (p.  376),  that  the  attrac- 
tion existing  between  the  planet  and  the  satellite  is  proportional 
to  the  product  of  their  masses,  and  inversely  proportional  to  the 
square  of  the  orbital  radius.  If  we  let  M  indicate  the  planet's  mass, 
and  m  that  of  the  satellite,  this  force  is  : 

n  Mm 
fr—r-'j 

r2 

where  G  is  a  constant  depending  on  the  units  adopted  for  linear 
distances,  etc. 

Now  this  Newtonian  force  produces  the  acceleration  /  in  the 
planetary  mass  m  ;  and  since  force  is  measured  by  the  acceleration 
produced,  multiplied  by  the  mass  moved,  it  follows  that  : 


f      r 

J   =  tr— 

r2 
405 


ASTRONOMY 

and  this  is  the  acceleration  due  to  the  attraction  of  the  planet  on 
the  satellite. 

In  an  exactly  similar  way,  we  can  show  that  the  satellite  pro- 
duces an  acceleration  of  the  planet  equal  to  : 


so  that  the  total  acceleration  existing  between  the  two  bodies  is  : 


If  we  now  equate  this  value  of  the  acceleration  to  that  given 
in  the  equation  for  /,  we  have  : 

=        r 


r2  t2 

or  : 


Let  us  next  apply  this  equation  to  two  planets,  each  having  a 
satellite,  and  indicate  by  subscript  numbers  quantities  belonging 
to  the  two  planets.  We  thus  easily  obtain  the  proportion  : 


or: 


With  the  help  of  this  general  proportion,  we  can  now  find  the 
planet's  mass  as  compared  with  that  of  the  earth.  We  need  only 
let  the  subscript  1  refer  to  the  earth  and  moon,  and  the  subscript  2 
to  the  planet  and  satellite.  Then  everything  is  known  in  the 
proportion  except  M2  +  -w2,  if  we  have  determined  by  direct 
observation  the  distance  and  period  of  the  satellite  with  respect  to 
its  planet.  It  is  to  be  noted  that  this  method  gives  only  the  sum 
of  the  masses  of  the  planet  and  its  satellite,  not  the  mass  of  the 
planet  alone.  But  this  is  of  minor  importance,  since  the  satellites 

406 


APPENDIX 

are  almost  always  very  small  compared  with  their  planets :  and, 
in  any  case,  it  is  the  combined  mass  of  the  system,  including  both 
planet  and  satellite,  that  we  really  need  to  know.  For  it  is  this 
combined  mass  which  pulls  upon  other  bodies  in  space ;  and  it  is 
the  pulling  force  upon  such  other  bodies  which  must  be  used  in 
any  further  calculations  relating  to  orbits,  etc. 

When  a  planet  has  no  satellite,  as  in  the  case  of  Venus  and  Mer- 
cury, we  cannot  employ  the  above  simple  and  accurate  method. 
We  must  then  have  recourse  to  a  mathematical  discussion  of  the 
slight  perturbations  the  planet  produces  in  the  observed  motions 
of  other  bodies  in  the  solar  system.  These  perturbations,  of  course, 
depend  on  the  planet's  mass,  being  greater  for  a  massive  planet 
than  for  a  small  one ;  and  consequently  the  planetary  masses  must 
admit  of  numerical  evaluation  from  the  observed  perturbative 
effects  they  produce.  Unfortunately  our  knowledge  of  the  mass 
of  Mercury  is  still  incomplete ;  that  of  Venus,  however,  is  known 
with  some  precision. 

The  mass  of  a  planet  once  determined,  it  is  easy  to  calculate  the 
force  of  gravity  on  the  planet's  surface,  its  Superficial  Gravity, 
as  it  is  called.  If  we  designate  by  P  the  planet's  radius  in  terms 
of  the  earth's  radius,  and  by  g  the  planetary  superficial  gravity, 
analogous  to  the  customary  designation  of  the  force  of  gravity  on 
our  earth's  surface,  we  have  at  once,  from  Newton's  law  of  gravi- 
tation : 

a-M 
9~pi> 

where  M  is  the  planet's  mass  in  terms  of  the  earth's  mass. 

To  ascertain  the  planet's  density  in  comparison  with  that  of 
our  earth,  we  proceed  thus  :  We  know,  in  general : 

Mass  =  Volume  X  Density. 
Therefore  we  have  for  the  earth's  mass  Me : 

Me  =  FA, 

where  Ae  represents  the  terrestrial  density,  and  V,  the  earth's 
volume. 

And  for  the  planet  we  have  : 

MP  =  FA, 

407 


ASTRONOMY 

Consequently  : 

APFP  =  Mv 

A.F.     M.' 
But,  again  using  P  to  indicate  the  planet's  radius  : 

17 

L£  _    D3 

F. 

Therefore,  if  we  take  the  mass  of  the  earth  as  unity  : 


If  we  wish  the  actual  specific  gravity  of  the  planet,  compared 
with  water,  we  must  substitute  for  the  Ae  the  value  5.53,  as  de- 
termined by  means  of  the  Cavendish  experiment  (p.  110). 

Note  27.     Synodic  and  Sidereal  Periods  (p.  209). 

Let  us  indicate  by  /sld  and  EsiA  the  sidereal  periods  of  Jupiter 
and  the  earth,  each  expressed  in  mean  solar  days.  EM,  for  in- 
stance, is  then  365J,  approximately.  Then,  regarding  both  orbits 
as  circular,  and  the  motions  uniform,  the  earth  in  one  day  will  pass 

360° 
over  a  fraction  of  its  total  orbit  represented  by  —  —  ;  and  Jupiter 

360° 
will  pass  over  a  fraction  represented  by  —  —  •    These  two  fractions 

«Aid 

are  not  equal  :  if  we  take  the  difference  : 

360°     360° 


this  quantity  will  be  the  angle  by  which  the  earth  and  Jupiter 
fail  to  lie  in  a  straight  line,  as  seen  from  the  sun  at  the  end  of  one 
day  after  the  beginning  of  Jupiter's  synodic  year  (see  Fig.  55, 
p.  208). 

This  quantity  is  therefore,  by  definition,  Jupiter's  daily  synodic 
motion.     But  if  Jupiter's  entire  synodic  period  be  represented 

360° 
by  Jsyn,  its  daily  synodic  motion  will  also  be Equating  this 

"  syn 

with  the  above  value  of  the  same  quantity,  we  have  : 
360°     360°      360° . 

T  ~    1?  T        ' 

«/syn          ^sid  "sid 

408 


APPENDIX 

or:  1          1          1 

«^syn        -^sid        «/sid 

By  means  of  this  equation,  Jupiter's  synodic  period  may  be 
calculated  from  his  sidereal  period,  and  vice  versa;  for  EM  is 
known  to  be  365J  days. 

Note  28.     Periods  of  Inferior  Planet  (p.  210). 

The  synodic  motion,  as  in  the  case  of  a  superior  planet,  again 
depends  on  the  earth's  orbital  motion  as  well  as  on  that  of  the 
planet.  As  before,  the  daily  sidereal  motions  of  Venus  and  the 

S60°  S60° 

earth  may  be  represented  by  —  —  and  —  —    The  difference  will  be 

Vsid  #sid 

the  daily  synodic  motion  of  Venus,  supposed  seen  from  the  sun. 
This  quantity  is  : 

360°     360° 

F8ld      Em  ' 

Thus  the  formula  for  the  daily  synodic  motion  of  an  inferior 
planet  is  perfectly  analogous  to  that  for  a  superior  planet,  except 
that  the  terms  are  now  interchanged.  This  is  of  course  due  to 
the  fact  that  the  superior  planet  has  a  slower  angular  motion 
around  the  sun  than  the  earth,  while  the  inferior  planet  has  a 
faster  angular  motion.  But,  as  before,  if  F8yn  be  the  synodic 

o/^r\o 

period  of  Venus,  the  daily  synodic  motion  will  be  -  ;   and  we 

'syn 

have  :  360°      360°     360° 


il 


It  follows  that  for  any  planet  whatever  the  reciprocal  of  the 
synodic  period  is  always  equal  to  the  difference  between  the 
reciprocals  of  the  planet's  sidereal  period  and  the  earth's,  sidereal 
period  of  365J  days. 

Note  29.     Table  of  Periods  (p.  211). 

It  will  be  of  interest  to  calculate  some  of  the  numbers  in  the 
table  (p.  211)  by  means  of  the  formulas  in  Notes  27  and  28.  We 
find: 

409 


ASTRONOMY 


RECIPROCAL 
OF  SIDEREAL 
PERIOD 

RECIPROCAL 
OF  EARTH'S 
SIDEREAL 
PERIOD 

DIFFERENCE 

SYNODIC 
PERIOD 

Mercury  

.011364 

002738 

008626 

116 

.001456 

002738 

001282 

780 

Uranus 

000033 

002738 

002705 

370 

In  computing  the  numbers  in  this  table,  all  periods  have  been 
reduced  to  days ;  and  the  numbers  in  the  last  column  are  recipro- 
cals of  those  in  the  column  headed  "  Difference."  It  is  at  once  clear 
from  this  little  calculation  how  the  peculiarities  of  the  table  of 
periods  arise.  As  the  sidereal  periods  of  the  outer  planets  increase, 
the  reciprocals  of  these  periods  must  diminish.  Consequently, 
these  reciprocals  must  gradually  approach  zero,  and  the  numbers  in 
the  column  "  Difference  "  must  approach  the  value  .002738.  So  the 
numbers  in  the  final  column  of  synodic  periods  must  approach  the 
value  365  J,  or  the  earth's  period.  This  is  just  what  we  should 
expect.  For  the  outermost  planets  remain  practically  stationary 
for  many  days  among  the  fixed  stars,  and  must  therefore  have  a 
conjunction  every  time  the  earth  goes  around  its  orbit,  or  very 
nearly  so.  The  effect  of  their  own  slow  orbital  motion  on  their 
synodic  motion  is  necessarily  very  slight. 

Note  30.     Greatest  Elongation,  Mercury  and  Venus  (p.  212). 

In  Fig.  58  (p.  212)  the  triangle  SEV  is  right-angled  at  V.  We 
can  therefore  calculate  the  angle  SEV,  which  is  the  required  greatest 
elongation  angle,  by  means  of  the  formula : 


sin  SEV 


sin  of  greatest  elongation 


SE' 

distance  of  planet  from  sun 
distance  of  earth  from  sun 


Let  us  make  the  calculation  for  Mercury.  The  orbit  of  this 
planet  is  more  flattened  than  any  other  in  the  solar  system : 
the  approximate  distance  of  Mercury  from  the  sun  varies  from  28.5 

410 


APPENDIX 


to  43.5  million  miles.  Obviously,  the  greatest  elongation  will  be 
larger  if  it  happens  when  the  planet  is  in  that  part  of  its  orbit  which 
is  farthest  from  the  sun.  We  shall  therefore  make  the  calculation 
twice,  using  the  two  values  just  given  for  the  distance  from  Mercury 
to  the  sun.  We  have  : 


LEAST 

GREATEST 

Distance  of  Mercury 

285 

435 

Distance  of  earth 

930 

930 

1  4548 

1  6385 

1  9685 

1  9685 

Log  sin  greatest  elongation 

9  4863 

9  6700 

Greatest  elongation 

17°51' 

27°53' 

From  this  calculation  we  see  that  Mercury  can  never  attain 
an  angular  distance  from  the  sun  greater  than  28°,  as  seen  projected 
on  the  sky  from  the  earth ;  and  ordinarily  its  greatest  elongation 
will  be  much  less  than  28°. 

Note  31.     Temperature  of  Mars  (p.  226). 

The  distance  from  Mars  to  the  sun  is  about  1J  times  that  from 
the  earth  to  the  sun.  Therefore,  if  we  assume  the  heat  radiated 
by  the  sun  to  diminish  with  the  square  of  the  distance,  Mars 
1 


receives  only 


as  much  heat  as  the  earth,  or  f  as  much.    We 


(1.5)' 

may  also  assume  that,  on  the  average,  all  planets  radiate  annually 
the  same  amount  of  heat  they  receive  ;  otherwise  they  would  be- 
come continuously  hotter  or  colder.  Now  we  have  a  law  of  physics 
known  as  Stefan's  law,  which  gives  us  an  estimate  of  the  quantity 
of  heat  a  body  will  radiate  at  different  temperatures.  According 
to  this  law,  calling  the  quantity  of  radiated  heat  Q,  and  the  tem- 
perature F  (Fahrenheit),  we  have: 

for  the  earth,  Qe  =  (458°  +  Fe)4, 
for  Mars,         Qm  =  (458°  +  ^J4. 
But  if  each  planet  radiates  as  much  heat  as  it  receives, 


Q.     4 

411 


ASTRONOMY 

Therefore  :  (458°  +  FeY  =  9 

(458°  +  Fmy     4' 
Now  for  the  average  temperature  of  the  earth,  we  may  put 

F.  =  60°. 
Therefore  : 


458°  +  Fm  =  -V$  (518°)  =  0.82  X  518°  =  425°. 
So  that  : 

Fm  =  -  33°  Fahrenheit. 

This  result  is  of  course  uncertain,  because  we  cannot  be  sure 
that  Stefan's  law  is  really  reliable  in  the  case  of  Mars  and  the 
earth.  It  has  been  tested  in  the  laboratory  only,  and  for  a  black 
body  radiating  its  heat  freely. 

Note  32.     Saturn's  Ring  (p.  245). 

We  have  already  found  (p.  402)  a  formula  for  the  accelera- 
tion toward  the  center  of  an  orbit.  It  is  : 

/=<•— 

But,  according  to  Newton's  law,  /  is  inversely  proportional 
to  r2  ;  so  that  F2  must  be  inversely  proportional  to  r.  Therefore 
if  the  rings  are  really  a  mass  of  satellites,  the  squares  of  their 
linear  velocities  are  inversely  proportional  to  their  distances  from 
the  planet.  In  other  words,  the  outside  of  the  ring  should  revolve 
more  slowly  than  the  inside. 

The  outside  radius  of  the  ring  has  been  measured  by  the  usual 
methods  (p.  203)  to  be  86,500  miles  ;  the  inner,  55,700.  The  square 
roots  of  these  numbers  are  in  the  ratio  of  1  to  1.24  ;  while  the  ob- 
served linear  velocities  are  in  the  ratio  of  1  to  1.25.  There  is 
therefore  a  surprisingly  close  agreement  ;  and  there  can  be  no  doubt 
that  the  various  parts  of  the  rings  rotate  in  accordance  with 
Kepler's  harmonic  law,  and'  are  composed  of  satellite  swarms. 

Note  33.     Halley's  Transit  of  Venus  Method  (p.  269). 

We  must  first  show  how  to  calculate  the  length  of  the  chord  in 
seconds  of  arc.  In  Fig.  119,  let  S,  Vi,  and  E  be  the  positions  of  the 

412 


APPENDIX 

sun,  Venus,  and  the  earth  at  the  moment  of  inferior  conjunc- 
tion. Let  P  be  the  synodic  period  (p.  208)  of  Venus,  in  days. 
Then  Venus  gains  a  whole  revolution  of  360°  on  the  earth 
in  P  days,  from  the  defini- 
tion of  the  synodic  period. 
In  one  day  Venus  gains 

?S£.     Therefore,  if  we  let    s — ====    $; ^^^ 

FIG.  119.     Halley's  Method. 

the  arc  ViV2  represent  the 

synodic  gain  of  Venus  on  the  earth  in  a  day,  as  seen  from  the 

sun,  we  have: 

Angle  S  =  — —  • 

But  in  the  plane  triangle  SEVi,  we  have  : 

sin  S:sinE  =  V2E:VZS, 

since  the  sines  of  the  angles  of  any  plane  triangle  are  proportional 
to  the  opposite  sides. 
Therefore:  sin  £  =  M  sin  S. 

V2E 

V  Sf 
But  the  ratio  ~—  is  known  from  the  known  relative  lengths  of 


the  radii  of  the  two  orbits  belonging  to  Venus  and  the  earth  (cf.  p. 
262).  The  angle  S  being  also  known,  as  has  just  been  shown,  it 
follows  that  we  can  calculate  the  angle  E,  which  is  the  angular 
distance  through  which  Venus  advances  across  the  face  of  the  sun 
in  one  day,  as  seen  from  the  earth.  This  angle  is  transformed  into 
seconds  of  arc ;  and  the  observers  having  found  the  fraction  of  a 
day  required  by  Venus  to  traverse  the  observed  chords,  we  find 
at  once  by  proportion  the  lengths  of  the  chords  in  seconds  of  arc. 
As  soon  as  the  lengths  of  the  two  chords  SP  and  sp  (Fig.  73, 
p.  269)  thus  become  known  in  seconds  of  arc,  the  further  proceedings 
are  simple.  For  the  angular  semi-diameter,  or  radius,  of  the  sun's 
disk  is  of  course  known  also  in  seconds  of  arc  (p.  118);  conse- 
quently, it  is  possible  to  calculate  the  distances  Sa  and  Sb  (Fig.  73) 
in  seconds  of  arc,  and  also  their  difference  ab.  We  also  know 
(Fig.  73)  the  ratio  of  the  lines  VA  and  Fa,  because  we  know 

413 


ASTRONOMY 


the  relative  distances  of  Venus  and  the  earth  from  the  sun.  Va  is 
0.723  if  Aa  is  1.000.  Therefore  : 

Va :  VA  =  723  :  277 ; 

and  ab,  in  miles,  is  ff-f  AB,  provided,  of  course,  that  the  distance 
AB  is  perpendicular  to  the  plane  of  Venus'  orbit.  If  not,  it  is 
easy  to  calculate  the  necessary  correction.  Now,  knowing 

ab,  on  the  sun,  both  in 
miles  and  in  seconds 
of  arc  as  seen  from 
the  earth,  we  easily 

FIG.  120.    Haiiey's  Method.  obtain  the  distance  of 

the  sun.     The  simple 

Fig.  120  shows  how  this  is  done.  Calling  r  the  radius  of  the  earth's 
orbit,  or  the  distance  from  the  earth  to  the  sun,  we  have,  from  the 
right-angled  triangle  Eab,  in  which  the  line  ab  is  on  the  sun,  as 
usual : 

tan  a6(seconds  of  arc)  = 


or 


"(miles)  ' 

•(miles)  =        afrMM      . 
tan  ab  (seconds) 


Note  34.    Solar  Parallax  from  the  Aberration  of  Light  (p.  271). 

Let  us  study  somewhat  in  detail 
the  action  of  light  aberration.  In 
Fig.  121,  suppose  that  an  observer  at 
t  has  his  telescope  pointed  in  the 
direction  tT ;  that  the  earth,  carry- 
ing the  observer  and  telescope,  is  for 
the  moment  moving  in  its  annual 
orbit  in  the  direction  ttr,  with  the 
velocity  v  miles  per  second.  Now 
suppose  light  from  a  star  at  S  reaches 
T  at  the  moment  when  the  telescope 
is  in  the  position  tT.  And  suppose 
this  light  travels  with  a  velocity  of  V 
miles  per  second  in  the  direction  ST. 

Now  indicate  by  «  the  angle  tTt' '.     Then  we  may  say,  as  it  were, 

414 


FIG.  121.     Solar  Parallax  from 
Aberration  of  Light. 


APPENDIX 

that  if  the  velocities  v  and  V  are  properly  proportioned  to  fit  the 
angle  a,  the  light  will  "stay  in  the  telescope  tube"  while  the  tube 
is  moving  from  tT  to  t'T1 '.  We  shall  then  have : 

tan  «  =  |. 

This  equation  signifies  that  a  star  at  S  will  really  appear  pro- 
jected on  the  sky  in  the  direction  t'T'.  In  other  words,  the  aberra- 
tion of  light  displaces  the  apparent  position  of  the  star  on  the  sky 
through  the  angle  a. 

And  there  is  no  difficulty  in  measuring  this  angle  a:  for  the 
displacement  of  the  star  is  always  in  the  direction  of  the  earth's 
motion,  here  tt'.  And  as  that  motion  takes  place  in  a  nearly 
circular  orbit,  the  displacement  a  must  be  in  opposite  directions 
at  intervals  of  half  a  year  (cf.  p.  137).  For  the  earth's  orbital 
motion  is,  of  course,  reversed  in  direction  at  opposite  points  of 
the  orbit.  We  have  therefore  merely  to  determine  by  observation 
the  apparent  declination  of  a  star  on  the  sky  at  intervals  of  six 
months.  If  a  suitably  located  star  is  selected,  the  declination  will 
be  found  to  vary  by  twice  the  angle  a ;  about  41"  of  arc- 

From  this  we  easily  compute  the  solar  distance.  For  the 
velocity  of  light,  V,  is  known  from  laboratory  experiments.  With 
V  and  a  both  known,  we  can  compute  v  with  the  equation  just  ob- 
tained, and  v  is  the  earth's  linear  velocity  in  its  orbit.  Thus  it 
has  been  found  that  v  is  about  18.5  miles  per  second.  This  we  have 
now  to  multiply  by  the  number  of  seconds  in  a  year,  to  get  the 
linear  circumference  of  the  earth's  orbit.  Finally,  dividing  by 
2  TT,  we  have  the  orbital  radius,  or  the  solar  distance. 

Note  35.     Sun's  Mass  (p.  291). 

To  ascertain  this  quantity,  we  resume  the  formula  which  ex- 
presses the  acceleration  which  the  sun  gives  a  planet.  It  is 
(p.  402): 

s  =Y1  —  (velocity  of  planet  in  orbit)2  _ 

r          radius  of  planetary  orbit 

In  the  case  of  the  earth  r  is  93,000,000  miles.     Assuming  the  orbit 
approximately  circular,  we  can  find  its  circumference  by  the  formula 
Circumference  =  2  irr  ; 
415 


ASTRONOMY 

and  this  being  divided  by  the  number  of  seconds  in  a  sidereal 
year,  we  find  V,  the  linear  orbital  velocity  of  the  earth,  in  miles 
per  second.  It  is  approximately  18J.  Now,  calculating/,  we  find  : 

/  =  0.233  inch. 

If  we  now  let  g,  as  usual,  represent  the  constant  of  terrestrial  grav- 
ity, we  may  write  a  simple  proportion  by  the  aid  of  Newton's  law  : 

/..     _      sun's  mass       .     earth's  mass 
(sun's  distance)2  '  (earth's  radius)2 

This  proportion  is  a  direct  consequence  of  Newton's  law,  which 
makes  attractive  forces  proportional  to  masses,  and  inversely 
proportional  to  squares  of  distances.  The  earth's  radius  becomes 
the  distance  for  terrestrial  gravity  </,  because  the  earth  attracts  as 
if  its  mass  were  concentrated  at  its  center;  and  the  radius  is 
the  distance  from  the  center  to  the  surface,  where  gravity  acts. 

In  the  proportion  everything  is  known  but  the  solar  mass  :  we 
can  therefore  readily  calculate  it. 

Note  36.     Angle  at  Earth's  Center  for  Possible  Eclipse  (p.  300). 

To  find  the  size  of  the  angle  M\cS  in  Fig.  84,  we  consider  the  tri- 
angle O'MiC,  taking  the  point  MI  as  the  point  of  tangency  of  the 
moon  at  MI  with  the  line  O'O.  Then,  in  the  triangle  O'MiC  : 

sin  MM'  =  M10f 

sin  MiO'c      Mic 

But,  as  the  sines  of  these  small  angles  are  proportional  to  the  angles 
themselves,  we  may  write  : 


But          MiO'  =  O'O  -  MiO  =  93,000,000  -  240,000,  very  nearly  ; 
240,000. 
93000000  -  240000 


M£)'c  240000 

But         MiO'c  =  solar  parallax  =  8".8. 

MicQ'  =  8".8  X  386  =  57'  ; 

also  O'cS  =  sun's  angular  radius  as  seen  from  the  earth 

=  16',  approximately. 
416 


APPENDIX 

Therefore : 

MicS  =  MtfO'  +  O'cS  =  57'  +  16'=  1|°,  approximately. 

And  if  we  now  consider  Mi  to  be  at  the  center  of  the  moon,  the  angle 
MicS  will  be  increased  by  the  moon's  angular  radius  as  seen  from 
the  earth,  or  16'.  So  that,  finally,  the  angle  at  c  between  the 
centers  of  the  sun  and  the  moon  at  MI  is  1  J°  +  16',  or  1J°,  approxi- 
mately. 

Note  37.     Draconitic  Period  (p.  305). 

We  have  seen  (p.  299)  that  the  moon's  node  makes  a  complete 
circuit  of  the  ecliptic  in  19  years.  Therefore,  in  one  year  it  moves 

o  fin0  1  8  £*° 

— ,  or  18.5°.     In  one  month  it  will  move  about  ^^-,   or   1.54°. 
iy  \  2. 

The  moon  itself  moves  13°  per  day,  as  a  result  of  its  orbital  motion 

13° 

around  the  earth.     Therefore  it  will  move  - — ,  or  0.54°  per  hour. 

So  the  moon  will  require  about  -^— -  hours,  or  about  three  hours, 

.OT: 

to  move  the  distance  traveled  by  the  lunar  node  in  a  month. 
Hence  the  difference  of  three  hours  between  the  draconitic  and 
sidereal  lunar  periods. 

Note  38.     Stellar  Magnitudes  (p.  324). 

It  is  possible  to  express  the  light-ratio  relations  by  means  of 
very  simple  formulas. 

Let  M\,Ni  be  the  brightness,  or  luminosity,  of  stars  of  the  mth  and 
nth  magnitudes ;  and  let  n  be  the  larger  number,  belonging  to 
the  fainter  star. 

Then:  M 

N 

or,  passing  to  logarithms  : 

n  -  m)  =  0.4(n  -  m). 


N 

From  this  we  also  obtain : 


M 

n  -  m  =  2.5  log  4 

N 


2E  417 


ASTRONOMY 

These  two  equations  enable  us  to  calculate  the  light-ratio  from 
the  difference  of  magnitudes,  and  vice  versa. 

Note  39.     Stellar  Photometry  (p.  325). 

To  understand  how  this  is  done,  we  shall  first  consider  the  fol- 
lowing interesting  question.  What  are  the  faintest  stars  that  can 
be  seen  with  a  telescope  of  given  size  ?  The  answer  here  depends 
on  the  diameter  of  the  object-glass,  because  this  determines  its 
area ;  and  the  area,  or  light-gathering  surface,  in  turn  determines 
the  light-gathering  power.  Now  it  has  been  found,  by  experiment, 
that  the  faintest  star  visible  in  a  telescope  having  an  object-glass 
one  inch  in  diameter  is  of  the  ninth  magnitude.  An  object-glass 
of  diameter  d  inches  will  have  an  area  dz  times  as  great,  and  will 
therefore  gather  d2  times  as  much  light.  Consequently,  it  will 
just  show  a  star  sending  us  a  quantity  of  light  equal  to : 

the  light  of  a  ninth-magnitude  star 
d2 

If  we  assume  this  star  to  be  of  the  nth  magnitude,  we  can  apply 
the  last  equation  of  Note  38.  We  then  have,  putting  m  =  9  : 

M  =  light  of  a  ninth-magnitude  star, 
*r  _  light  of  a  ninth-magnitude  start 

And  then  our  equation  gives  : 

n-  9  =  2.51og^=  2.5  log  d2; 

or,  n  =  9  +  2.5  log  d2. 

This  simple  equation  tells  us  the  magnitude  n,  of  a  star  just 
visible  in  a  telescope  of  which  the  object-glass  has  a  diameter  of 
d  inches.  And  it  also  enables  us  to  calculate  the  magnitude  of  a 
star  just  visible  through  a  diaphragm  of  which  the  aperture  simi- 
larly has  a  diameter  of  d  inches. 

Note  40.     Light  emitted  by  Vega  (p.  326). 

As  we  have  stated,  rough  measurements  show  that  the  entire 
quantity  of  starlight  received  by  an  observer  on  the  earth  is  equal 
to  that  of  2000  Vegas.  This  has  also  been  estimated  as  being 

418 


APPENDIX 

equivalent  to  -g-sWowo  °f  sunlight.     Therefore,  we  receive  from 

Vega  : 

sunlight  ...     1  sunlight 

33000000     2000'     r  66000000000' 

and  the  word  "  sunlight7'  here  means  the  quantity  of  light  received 
from  the  sun.  Then,  since  the  intensity  of  light  diminishes  pro- 
portionately to  the  square  of  our  distance  from  its  source,  Vega 
must  emit  : 

light  emitted  by  sun  y,  (distance  of  Vega)2 
66000000000  v  (distance  of  sun)2  ' 

But  Vega  is  one  of  the  stars  whose  distance  has  been  measured, 
approximately.     It  has  been  found  that  : 

Vega's  distance  =  1820000 
sun  s  distance 
Therefore,  Vega  must  emit  : 


. 

or,  approximately  :  Light  emitted  by  sun  X  49. 

Note  41.     Motion  of  Solar  System  (p.  338). 

Figure  122  may  make  this  matter  clearer.  The  solar  system  is 
for  the  moment  imagined  stationary,  and  the  stars  all  moving 
with  parallel  annual  velocities  represented  by  the  arrows  'SSi. 

Solar 

5^  S  O 

System 


FIG.  122.     Motion  of  Solar  System. 

•4 

On  each  of  these  arrows  a  parallelogram  is  constructed,  having 
one  side  SSit  directed  toward  the  solar  system,  or  away  from  it. 
In  the  two  parallelograms  shown  in  the  figure,  the  diagonal  velocity 
SSi  may  be  regarded  as  equivalent  to,  and  it  may  be  replaced  by, 
the  two  smaller  velocity  arrows  forming  the  sides  of  the  parallelo- 

419 


ASTRONOMY 

grams  (cf.  p.  399).  Only  the  part  SSi  affects  the  velocity  of  ap- 
proach or  recession  with  respect  to  the  solar  system.  The  entire 
arrow  SSi  indicates  approach  on  the  right-hand  side  of  the  figure, 
and  recession  on  the  left-hand  side.  At  the  lower  edge  of  the 
diagram  appears  a  star  none  of  whose  real  velocity  SSi  will  appear 
as  either  approach  or  recession. 

We  might  satisfy  the  above  observations  if  all  the  arrows  SSi 
were  replaced  by  a  single  parallel  arrow,  starting  from  the  solar 
system,  and  pointing  toward  the  right.  A  study  of  radial  veloci- 
ties all  around  the  sky  must  therefore  prove  one  of  two  things : 
either  a  stream  of  stars  is  passing  us  in  a  definite  direction,  or  the 
solar  system  is  moving  with  an  equal  velocity  in  the  opposite  direc- 
tion. The  latter  hypothesis  is,  of  course,  the  more  probable. 

Note  42.     Distance  of  Vega  (p.  341). 

Figure  123  shows  the  sun,  the  earth,  and  Vega.     The  parallax 

angle,  0".ll,  is, 
by  definition,  the 
angle  EVS,  sub- 
tended at 


by  the  radius  of 

FIG.  123.     Distance  of  Vega.  , ,  , ,  ,  ,  ., 

the  earth  s   orbit 

around  the  sun.     As  usual,  we  can  solve  the  right-angled  triangle 
ESV,  in  which  we  know  the  angle  at  V  and  the  side  ES.    We  have  : 

tanO".ll=f|,or,Sy  =       ES 


SV  tan  0".ll 

But  tan  0".ll  is,  approximately  : 

0.11 
200000  ' 

and  so  SV,  the  distance  of  Vega,  is : 

93000000  X  2000QO 
0.11 

Note  43.     Mass  of  Binary  Star  (p.  350). 

Referring  to  Note  261  (p.  405),  the  formula  in  the  case  of  a  binary 

system  is:  Mass  of  system  =  f^, 

O  t 

420 


APPENDIX 


where  S  is  the  sun's  mass,  a  the  linear  diameter  of  the  binary 
orbit  in  terms  of  the  distance  earth-to-sun  as  a  unit,  and  t  the 
binary  orbital  period  in  years. 

Note  44.     Size  of  Andromeda  Nebula  (p.  353). 

Figure  124  will  make  this  clear.  S  is  the  sun ;  E,  the  earth ;  and 
Nj  the  center  of  the  nebula.  C  and  C"  are  points  on  the  circumfer- 
ence of  the  nebula.  The 
angle  ENS  is  the  paral- 
lax of  the  nebula,  here 
assumed  to  be  0".01 ; 
since  it  is  by  definition 
the  angle  subtended  by 
ESj  the  radius  of  the 
earth's  orbit,  to  a  sup- 
posed observer  at  the 
nebula.  The  angle  CSC'  is  the  angular  diameter  of  the  nebula, 
seen  from  the  solar  system,  and  it  is  1.5°.  Therefore  the  linear 
distance  CC'  must  be  greater  than  ES  in  the  approximate  ratio 
1.5° 


FIG.  124.     Size  of  Andromeda  Nebula. 


0".01 


,  or  540,000. 


Note  45.     Attraction  of  Andromeda  Nebula  (p.  354). 

Regarding  the  nebula  as  approximately  a  globe  540,000 
X  93,000,000  miles  in  diameter,  and  the  sun  as  a  globe  1,000,000 
miles  in  diameter,  the  volume  of  the  nebula  equals  (540,000  X  93)3 
times  the  sun's  volume.  Let  us  imagine  the  sun  and  nebula  to 
have  equal  densities.  Then  their  masses  will  be  in  the  above  ratio 
of  their  volumes.  But  with  a  parallax  of  0".01,  the  nebula  is 
20,000,000  times  as  far  away  from  us  as  the  sun.  Therefore  the 
relative  attractions  of  nebula  and  sun  on  the  earth  are : 


(540000  X  93)' 


,  or,  approximately,  310000000. 


(20000000)2 

It  follows  that  the  nebular  density  may  be  as  slight  as  SIOOOQOOIT 
of  the  solar  density,  and  yet  the  earth  be  attracted  by  the  neb- 
ula as  much  as  by  the  sun. 


421 


INDEX 


Aberration  of  light, 

explanation  of,  136 

solar  parallax  from,  271,  414 
Absolute  method  of  measurement,  331 
Absorption, 

of  light  by  gases,  283 
in  the  sun,  573 

of  starlight  by  atmosphere,  286 
Adams,  discovers  Neptune,  248 
Aerolites,  320 

cosmic  dust,  320 

Afternoon,  not  equal  to  morning,  135 
Airy,  Sir  Geo.,  astronomer  royal,  31 
Albedo,  218 

of  Mercury,  218 

of  Venus,  220 
Aldebaran, 

standard  first-magnitude  star,  324 
Alexandria,  Eratosthenes  measures  earth 

there,  94 

Algol,  variable  star,  328 
Almanac, 

nautical,  155,  202 

use  in  finding  planets,  50 
Altitude,  36 

American  Ephemeris,  202 
Andromeda,  constellation, 

nebula  in,  353 
size  of,  421 
Angle,  defined,  29 
Angular  diameter, 

of  moon,  173 

of  planets,  203 

of  sun,  118 

Angular  distance,  defined,  29 
Annular  eclipses  of  sun,  304 
Apex  of  sun's  motion  in  space,  338 
Aphelion,  263 
Apogee,  tides  at,  254 
Apollo,  ancient  name  of  planet  Mercury, 

217 

Apparent  orbit,  of  binary  stars,  348 
Apparent  solar  day,  65 

variable  in  length,  71 
Apparent  solar  time,  65 

explanation  of,  67 


Apparent  solar  time,  table  of  differences 

from  mean  solar  time,  82 
Arago,  compliments  Herschel,  247 
Areas,  Kepler's  law  of,  120,  184,  399 
Aristarchus  of  Samos, 

his  solstice  observation  used  by  Hip- 

parchus,  127 

Aristotle,  explains  lunar  phases,  163 
Aristyllus,  star  observations,  129 
Artificial     star,     used     in     photometry, 

"    325 
Ascension  island,  Gill's  Mars  expedition 

to,  266 

Aspect  of  heavens,  on  different  dates,  72 
Astronomer  royal, 

Airy,  31 

Bradley,  136 

office  established  by  Charles  II,   159 
Astronomy, 

Chinese,  discovery  of  ecliptic,  29 

definition  of  the  word,  2 

Greek,  30 

popular  questions  concerning,  2 

value  as  a  study,  21 

value  for  practical  purposes,  18 
Atmosphere, 

absence  of,  on  moon,  166 

extent  of  terrestrial,  from  meteor  ob- 
servations, 319 

light  absorption  by,  286 

of  earth,  113 

heats  meteors,  318 

of  Jupiter,  236 

of  Mars,  222 

of  Mercury,  218   - 

of  Venus,  220 

interferes  with  transit,  269 

produces  refraction,  114 

produces  twilight,  113 

retains  solar  heat,  113 
Auriga,  constellation,  diagram  of,  61 
Aurora,  periodicity  of,  290 
Average, 

distance  between  stars,  346 

stellar  statistics,  342 
Axial  rotation,  see  Rotation,  axial 


423 


INDEX 


Axis,  rotation, 

direction  in  space,  planets,  202 

sun,  296 

of  celestial  sphere,  32 
of  earth,  31 

of  equatorial  mounting,  279 
of  telescope  mounting,  276 

Balance,  torsion,  107 

constant  of,  108 

Base-line,  for  parallax  measures,  262 
Bayer,  observes  Castor  and  Pollux,  327 
Beard,  of  comets,  307 
Bessel,  measures  stellar  parallax,  192,  333 
Biela,  his  comet  breaks  up,  319 
Binary  stars,  see  Stars,  binary 
Blue,  color  of  sky,  113 
Bode's  law,  196 

in  case  of  Ceres,  232 
Bond,  discovers  satellite  of  Saturn,  246 
Bouguer,  measures  arc  in  Peru,  99 
Bradley,  J.,  astronomer  royal, 

discovers  aberration  of  light,  136 
Bright-line  spectrum,  283 
Bureau  of  Standards  at  Washington,  102 

Caesar,  his  calendar,  138 
Calendar,  the,  138 

ecclesiastical,  146,  385 

perpetual,  147 
Campbell,  determines  apex,  338 

observes  Mars,  224 
Canals,  of  Mars,  223 
Cape  of  Good  Hope  observatory, 

heliometer  at,  267 
Cassini,  computes  Horrocks'  observation, 

270 

Cassiopeia,  constellation,  how  to  find,  54 
Cavendish,  weighs  the  earth,  107,  376 
Cayenne,  Richer  swings  pendulum  at,  98 
Celestial   equator,    precessional   motion, 

129 
Celestial  meridian,  36 

correspondence  with  terrestrial,  73,  367 
Celestial  poles,  32 

motion  of,  seen  by  a  traveler,  39 

position  with  respect  to  horizon,  40, 

365 
Celestial  sphere,  24 

apparent  rotation  of,  30 

oblique,  42 

parallel,  41 

right,  40 
Center  of  gravity, 

binary  stars,  347 

earth  and  moon,  174 


Central, 

force,  in  planetary  motion,  187 

sun,  355 

Centrifugal  force  on  the  earth,  98 
Century,  number  in  calendar,  143 
Ceres,  the  first  planetoid,  232 
Chaldeans,  discover  the  Saros,  304 
Chamberlin,     planetesimal     hypothesis, 

358 

Charles  II,  establishes  office  of  astrono- 
mer royal,  159 
Charting  and  mapping,  19 
Chemistry, 

of  aerolites,  321 

of  sun,  286 

and  stars,  with  spectroscope,  284 

stellar,  by  Huggins,  337 
Chinese  astronomy,  discovery  of  ecliptic, 

29 

Chromosphere,  of  sun,  293 
Chronograph,  electric,  278 
Chronometer, 

marine,  used  in  navigation,  157 

earliest,  159 
Church      calendar,     see      Ecclesiastical 

calendar 
Circle, 

diurnal,  33 

ecliptic,  28 

graduated,  on  sextant,  152 
on  telescope,  278,  281 

great,  defined,  27 

meridian,  277 
Clerk-Maxwell, 

constitution  of  Saturn's  ring,  245 

light-pressure  theory,  309 
Clock, 

astronomic,  standard,  278 

of  equatorial  telescope,  280 

regulator,  jeweler's,  279 
Clusters  of  stars,  351 

distance  and  size,  352 

nebulous  matter  in,  352 
Coal-sack  in  Milky  Way,  354 
Collimator,  in  spectroscope,  282 
Collision, 

of  stars,  347 

possible  with  comet,  308 
Color  of  sky,  113 
Coma,  of  comets,  307 
Comets,  14,  307 

designation  of,  311 

capture  theory  of,  314 

light-pressure  theory  of  tails,  309 

periodic,  313 
Compound  lenses  of  telescope,  273 


424 


INDEX 


Conic  sections,  Newton's  comet  orbits,    Dead-reckoning,  in  navigation,  151 


312 
Conjunction,  209 

superior  and  inferior,  210 

of  Venus,  produces  transit,  268 
Conservation  of  energy,  2 

effect  on  tidal  friction,  256 
Constant, 

of  torsion  balance,  108 

in  calendar  calculations,  144 
Constellations,  7 

diagrams  of  principal,  63 
Continuous  spectrum,  283 
Cooling  of  stars,  6 
Copernicus, 

his  book  De  Revolutionibus,  87 

his  planetary  theory,  191 
Cordova  catalogue  of  stars,  334 
Corona,  solar,  295 

Correction,  of  sextant  observations,  156 
Cosmic  dust,  320 
Cosmic  velocity, 

of  solar  system,  338 

of  stars,  346 
Cosmogony,  356 
Council  of  Nice,  146 
Crests,  of  tidal  waves,  253 
Cross-threads,  in  telescope,  275 
Crystal  sphere,  in  Ptolemaic  theory,  189 
Curvature,  of  earth, 

arguments  proving,  87 

measurement  of,  97 
Curves  in  planetary  motion,  215 
Cygnus,  constellation,  diagram  of,  61 

Darwin,  theory  of  moon's  origin,  258 
Date,  in  calendar, 

four  parts  of,  138 

of  Easter,  148 
Date-line,  international,  75 
Day,  65 

apparent  solar,  67 

lengthening  of,  by  tidal  frictions,  257 

lunar,  176 

midsummer  and  midwinter,  121 

on  Mars,  221 
Mercury,  218 
Venus,  221 

planetary,  202 

sidereal,  66 

solar,  unequal,  71 
Day  and  night,  31 

at  the  pole,  42 

equal  at  equator,  40 

in  temperate  regions,  43 

longest  and  shortest,  121 


Declination,  defined,  34,  363 

measured,  278 

Deferent,  in  Ptolemaic  theory,  189 
Degree  of  latitude,  varying  length,  97 
Deimos,  satellite  of  Mars,  222 
Demon  star,  see  Algol 
Density, 

comets,  308 

earth,  110 

moon,  175 

nebulae,  354 

stellar  distribution,  345 

sun,  292 

Departure,  in  navigation,  151 
Diamete.r, 

angular,  of  moon,  172 
sun,  118,  291 
planets,  203 

of  planets,  in  miles,  204 

of  sun,  in  miles,  291 
Differences  of  time,  72 

sidereal  and  solar,  73 
Differential  method  of  measurement,  332 
Dipper,  constellation,  see  Ursa  Major 
Disks,  planetary,  13 

seen  in  field  glass,  52 
Distance,  see  also  Parallax 

change  of  stellar,  observed  with  spec- 
troscope, 284 

of  star  clusters,  352 

of  stars,  322,  330 

of  Vega,  420 
Diurnal, 

circles,  33 

inequality  of  tides,  253 

observations  of  Mars  for  solar  parallax, 

263 

Doppler  principle,  with  spectroscope,  284 
Double 

stars,  8.     See  also  Stars,  binary 

telescopes,  photographic,  281 

double  star,  351 

Douglass,  observations  of  Mars,  227 
Draconitic  period,  305,  417 

Earth, 

an  astronomic  body,  15 
atmosphere  of,  113 
curvature  of,  87,  97 
density  of,  110 
flattening  of,  97 
interior  of,  111 
measurement  of,  92,  95 
orbit,  form  of,  116 
rotation,  15,  30,  88 


425 


INDEX 


Earth,  shadow  of,  in  eclipses,  303 

shape  a  geoid,  101 

weighing  it,  103 
Earth-shine  on  moon,  164 
Easter,  date  of,  148,  385 
Eccentricity  of  planetary  orbits,  200 
Ecclesiastical  calendar,  146 
Eclipses,  297 

annular,  of  sun,  304 

lunar,  301 

solar,  297 

periodicity  and  Saros,  304 

umbra  and  penumbra,  303 

variable  star,  328 
Ecliptic,  26 

locating  it  on  sky,  47 

pole  of,  131 

Egyptians,  find  length  of  year,  127 
Electric  chronograph,  278 
Elements, 

chemical,  in  sun,  284,  286 
in  aerolites,  321 

of  orbits,  binary  stars,  348 
comets,  313 
planets,  201 

perturbations,  206 
Elongation, 

maximum,  of  planets,  212,  410 

of  satellites  from  planets,  205 
Energy,  conservation  of,  2 

meteors,  318 

tidal  friction,  256 
Ephemeris,  planetary,  202 
Epicycle,  in  Ptolemaic  theory,  189 
Equation  of  time,  134 

table  of,  82 

Equation,  personal,  see  Personal  error 
Equator, 

terrestrial  and  celestial,  33 

processional  motion  of,  129 
Equatorial,  mounting  of  telescopes,  279 
Equinoxes,  35,  43,  364 

precession  of,  126 
Eratosthenes, 

measures  ecliptic,  30 
size  of  earth,  92 

observes  j8  Librae,  327 
Erecting  lens,  in  telescope,  273 
Eros,  planetoid,  its  orbit,  236 

observed  for  solar  parallax,  267 
Evolution,  tidal,  256 
Eye-piece,  in  telescope,  273 

Faculae,  of  sun,  289 

Fixed  stars,  see  Stars 

Flagstaff,  Lowell  observatory  at,  227 


Flash  spectrum,  in  solar  eclipse,  288 
Flattening, 

of  earth,  97 

of  planets,  205 

of  planets'  orbits,  200 
Focus, 

of  earth's  orbit,  116 

of  telescope,  272 
Force, 

centrifugal,  on  earth,  98 

central,  in  planetary  motion,  187 

tidal,  252 

repulsive,  of  sunlight,  309 
Foucault  experiment,  89,  371 
Fraunhofer,  spectrum  lines,  287,  308 1 
Friction,  tidal,  256 

Galaxy,  see  Milky  Way 
Galileo, 

advocates  rotation  of  earth,  88 

discovers  Jupiter's  satellites,  237 

his  telescope,  273 

observes  Saturn's  ring,  241 
sunspots,  17 

phases  of  Venus,  219 
Galle,  discovers  Neptune,  248 
Gauss, 

computes  orbit  of  Ceres,  233 

Easter  date,  148,  385 
Gemini,  constellation,  diagram  of,  61 
Geodesy,  95 
Geography, 

latitude  and  longitude  in,  34 

terrestrial  meridians  in,  73 
Geoid,  shape  of  earth,  101 
Georgium  Sidus, 

name  for  Uranus,  247 
Gill, 

observes  Mars  for  solar  parallax,  266 
Globe,  celestial,  37 

use  of,  63 
Gnomon,  of  sundial,  78 

construction  of,  368 

Gottingen,  observatory,  heliometer,  267 
Graduated  circle, 

of  sextant,  152 

of  telescope,  281 
Gravitation, 

action  of,  inside  nebulae,  4 

force  of,  on  sun,  102,  291 

Newton's  law  of,  103,  184 

proves  distance  of  stars,  322 
Great     Bear,     constellation,     see     Ursa 

Major 

Great  circle  of  sphere,  defined,  27 
Greatest  luminosity  of  Venus,  219 
426 


INDEX 


Greenwich,  initial  meridian,  34,  73 
Gregorian  calendar,  138 
Groom  bridge, 

catalogue  of  stars,  334 

his  runaway  star,  347 

Hale,  spectroheliograph,  294 
Halley, 

his  comet,  311 

transit  of  Venus  method,  269,  412 
Harmonic  law  of  Kepler,  188,  400 
Harvest  moon,  177 
Heat, 

interior  of  earth,  111 

of  meteors,  cause  of,  318 

of  stars,  326 

solar,  retained  by  atmosphere,  113 
Heavens,  aspect  on  different  dates,  72 
Height  of  meteors,  319 
Heliacal  rising  of  stars,  127 
Heliometer,  267 

Helmholtz,  theory  of  solar  energy,  292 
Hemisphere,  southern,  seasons,  122 
Herschel,  Captain  John,  perpetual  calen- 
dar, 147 
Herschel,  Sir  John, 

distances  in  solar  systems,  12 

star  magnitudes,  323 

i]  Argus  observation,  327 
Herschel,  Sir  William, 

apex  of  sun's  way,  339 

discovers  satellite  of  Saturn,  246 

discovers  satellite  of  Uranus,  247 

discovers  Uranus,  232 

explains  Galaxy,  354 

star  counts,  355 
Hipparchus, 

explains  eclipses,  297 

observes  solstice,  127 

originates  Ptolemaic  theory,  189 

scale  of  star  magnitudes,  323 
Honolulu  expedition,  112 
Hooke,  corresponds  with  Newton,  91 

ideas  about  comets,  312 
Horizon,  defined,  36 
Horns,  of  moon,  165 
Horrocks, 

observes  transit  of  Venus,  270 
Hottest  day  of  summer,  122 
Hour-angle, 

defined,  66,  363 

measures  time,  66 

relation  to  sidereal  time,  366 
Hour-circle,  defined,  363 
How  to  know  the  stars,  45 
Huggins,  spectroscopist,  336 


Hutton,  calculates  Maskelyne's  observa- 
tions, 104 
Huygens, 

Saturn's  ring,  13,  241 
Hydrocarbon,  in  comets,  308 
Hydrogen  jets  from  sun,  293 

Ice  age,  geologic,  125 

Image,  focal,  in  telescope,  272 

Imperfections  of  visual  observations,  228 

Inclination, 

of  lunar  orbit,  160,  298 

of  planetary  orbit,  200 
Inequality, 

diurnal,  of  tides,  253 

of  morning  and  afternoon,  135 
Inferior  planets,  209 

conjunctions,  210 

oscillations  of,  213 

period  of,  409 
Inhabitants  of  Mars,  223 
Instruments,  astronomic,  272 
International  date-line,  75 
Invariable  plane,  207 
Iris,  observed  by  Gill,  267 

Janssen,  observes  prominences,  293 
Julian  calendar,  138 
Juno,  planetoid,  234 
Jupiter, 

appearance  in  telescope,  13 

atmosphere,  236 

comet  family  of,  314 

distance  from  sun,  240 

how  to  find,  51 

influence  on  planetoid  orbits,  235 

in  nebular  hypothesis,  358 

longitude   from  satellite  observations, 
.  239 

markings  on,  236 

rotation,  axial,  236 

satellites,  237 

seasons  and  temperature,  237 

Kapteyn,  stellar  researches,  342 

Keeler, 

constitution  of  Saturn's  ring,  245 
planetesimal  hypothesis,  359 
spiral  nebulae,  353 

Kepler, 

ellipticity  of  earth's  orbit,  116 
harmonic  law,  188,  400 
ideas  about  comets,  312 
law  of  areas,  120,  184,  399 
laws  apply  to  binary  stars,  348 
variation   in   distance   from   earth   to 
sun,  193 


427 


INDEX 


Kinetic  theory  of  gases, 

explains  absence,  of  lunar  atmosphere, 

167 

of  Martian  atmosphere,  222 
in  nebular  hypothesis,  357 
similar  to  cosmic  stellar  theory,  347 
Ktistner,  latitude  variation,  112 

La  Condamine,  Peru  arc,  99 
Lagrange,  planetoid  orbits,  235 
Land-fall,  in  navigation,  151 
Laplace, 

capture  theory  of  comets,  314 

nebular  hypothesis,  235,  356 
binary  stars,  350 
satellites  of  Uranus,  247 
Lapland,  arc  measured  in,  99 
Lassell,  satellites  of  Uranus,  247 
Latitude, 

arcs  of,  used  in  <-  ^odesy,  99 

found  in  navigation,  154 

terrestrial,  34 

variation  of,  112 
Lava,  source  of,  112 
Law,  Bode's,  196 

Kepler's,  187,  399 

Newton's,  103,  184 
Layer,  reversing,  in  sun,  288 
Leap-year,  rule  for,  142 
Leipzig  observatory,  heliometer,  267 
Lens,  telescopic,  compound  and  erecting, 

273 

Leo,  constellation,  diagram  of,  62 
Leonid  meteors,  316 
Leverrier,  discovers  Neptune,  248 
Lexell,  explains  Uranus,  247 
Libration, 

of  moon,  171 

of  Mercury,  218 
Light, 

aberration  of,  136 

solar  parallax,  271,  414 

absorption  of,  by  gases,  283 
terrestrial  atmosphere,  325 

gathering  power  of  telescopes,  275 

of  meteors,  cause  of,  318 

of  sun,  source  of,  292 

total,  of  stars,  325 

velocity  of,  333 

Light-pressure  in  comet  tails,  309 
Light-ratio  of  star  magnitudes,  324 
Light-year  and  stellar  parallax,  333 
Limits,  in  eclipses,  300 
Line-of-sight, 

motions  of  stars  in,  334 

of  telescope,  277 


Lockyer,  observes  prominences,  293 
Logogriph  of  Huygens,  241   • 
Long  Island  Sound,  tides  in,  256 
Longitude, 

arcs  of,  used  in  geodesy,  99 
determined   from    Jupiter's   satellites, 

239 

occultations,  239 
in  navigation,  157 
terrestrial,  34,  73 
Lowell, 

markings  on  Mercury,  218 

Venus,  22; 
Mars  observations,  226 

Magnetic  storms,  periodicity  of,  290 
Magnifying  power  of  telescopes,  274 
Magnitudes  of  stars,  323 
Mapping  and  charting,  19 
Maps  of  stars,  45 
Markings, 

on  Mercury,  218 
Mars,  223 
Jupiter,  236 
Venus,  221 
Mars, 

how  to  find,  51 
inhabitants,  223 
observed  for  solar  parallax,  263 
Maskelyne,  weighs  the  earth,  104 
Mass, 

distinction  from  weight,  102 
of  Algol,  329 

binary  stars,  350,  420 
comets,  308 
earth,  110 
moon,  173,  396 
planets,  204,  405 
sun,  291,  415 
Matter,  2 

Maupertuis,  arc  in  Lapland,  99 
Maxwell,  J.  Clerk, 
light-pressure,  309 
Saturn's  ring,  245 
Mean  solar  day,  65,  71 
Mean  solar  time,  71 

table  of  difference  from  apparent  time, 

82 

Mercury,  217 
how  to  find,  50 
transits  of,  306 
Meridian, 

celestial,  36,  73 

circle,  276 

distance  of  stars  from,  67 

of  Greenwich,  34 


428 


INDEX 


Meridian, 

planets  on  it  at  midnight,  52 

right-ascension  of,  67,  366 

shape  of  terrestrial,  97 

standard,  74 
Meteors,  315 
Micrometer,  276 

used  for  binary  stars,  347 
Mars,  265 
stellar  parallax,  332 
Midsummer  and  midwinter  day,  121 
Milky  Way,  354 
Minor  planets,  see  Planetoids 
Mira,  variable  star,  328 
Month,  lunar,  in  eclipses,  298 
Moon, 

absence  of  atmosphere,  17,  166 

always  near  ecliptic,  49,  160 

angular  velocity  of  apparent  motion, 
161 

axial  rotation,  169 

causes  tides,  252 

craters  and  mountains,  17,  181 

Darwin's  theory  of  its  origin,  258 

density,  175 

dimensions,  16 

distance,  16,  169,  394 

draconitic  period,  305,  417 

earth-shine  on,  164 

eclipses,  164,  297 
limit,  300 
where  visible,  303 

effect  of  tides  on,  257 

harvest,  177 

horns  turn  away  from  sun,  165 

its  attraction  produces  precession,  130 

libration,  171 

lunar  day,  176 

measurement  of  mountains  on,  182 

measurement  of  size  of,  173 

month  of,  298 

not  self-luminous,  16,  160 

occults  stars,  166 

orbit  plane,  inclination  of,  160 

orbit  revolution,  16,  160 

our  nearest  neighbor,  16,  160 

parallax  of,  169,  396 

perigee  of,  169 

phases,  16,  162 
when  eclipsed,  301 

physical  appearance,  16 

rides  high,  179 

rising,  variation  in  time  of,  176 

shape  of  orbit,  168,  181,  397 

sidereal  and  synodic  periods,  161 

surface  features,  16 


volume,  173 

weight  or  mass,  173,  396 
Morning  and  afternoon  unequal,  135 
Morning  and  evening  stars,  22 
Moulton,  planetesimal  hypothesis,  358 
Mountain, 

used  to  weigh  the  earth,  104 

measurement  of  lunar,  182 
Mountings,  for  telescopes,  276 
Multiple  stars,  351 

Nautical  almanac,  155,  202 
Navigation, 

an  astronomic  process,  20 

dead-reckoning  in,  151 

"departure"  in,  151 

early  method  of,  159 

finding  ship's  latitude,  154 

fundamental  problem  of,  151 

nautical  almanac  in,  155 

sextant  in,  152 
Neap  tides,  255 
Nebula, 

density,  354 

effect  of  internal  gravitation,  4 

gaseous  constitution  of,  4 

in  Andromeda,  353,  421 

in  star  clusters,  352 

Laplace's  hypothesis,  235,  356 

nebulium  in,  353 

number  of,  5 

planetary,  352 

resolving  them,  3 

ring  form,  353 

spiral  form  predominant,  5,  353 

spiral,  in  planetesimal  hypothesis,  359 
Nebular  hypothesis,  235,  356 

satellites  of  Uranus  in,  247 
"Nebulium"  in  nebulae,  353 
Nebulosity  of  cometary  coma,  307 
Neptune,  planet,  247 
New  Haven  observatory,  267 
New  stars,  see  Temporary  stars 
Newton, 

comet  orbits,  312 

determines  flattening  of  earth,  98 

law  of  gravitation,  103,  184 

test  of  earth's  rotation,  91 
Nice,  council  of,  146 
Night,  radiation  from  earth  in,  122 
Nodes, 

motion  of  lunar,  in  eclipses,  299 

of  Milky  Way,  354 

planetary  orbital,  200 

transit  of  Venus  in,  268 
Nodules,  of  sun,  289 


429 


INDEX 


Novce,  see  Temporary  stars 
Nucleus, 

in  development  of  binary  stars,  350 

of  comets,  307 
Nutation,  of  terrestrial  axis,  132 

Object-glass,  of  telescope,  272 
Oblique  sphere,  42 
Observations, 

correction  of,  156 

imperfections  of  visual,  228 

planetary,  for  orbit  determination,  199 
Occultations,  of  stars,  166 

used  for  determining  longitude,  239 
Olbers,  telescopic  constellations,  310 
Opposition,  of  planets,  212 

of  Mars,  favorable,  263 
Orbit,  binary  star's,  347 

cometary,  312 

earth's,  around  sun,  25,  116 
slow  changes  of,  125 

eccentricity  of,  200 

elements  of,  six,  201 

inclination  of,  200 

meteoric,  316 

moon's,  shape  of,  168,  181,  397 

nodes,  line  of,  200 

perturbations  of,  206 

planetary,  197 

in  nebular  hypothesis,  356 

in  planetesimal  hypothesis,  358 

stability  of,  206 

of  planet's  satellites,  205 

stellar  parallactic,  331 
Orientation  of  sundial,  82 
Orion,  constellation,  diagram  of,  62 
Oscillations,  inferior  planets,  213 

tidal,  253 

Pallas,  planetoid,  233 

Pantheon,  Foucault  experiment,  89 

Parallax,  lunar,  169,  396 

solar,  aberration  of  light,  271,  414 

definition  of,  260 

definitive  value  of,  268 

diurnal  method,  263 

Gill's  observations,  266 

Mars  observations,  263 

perturbation  method,  271 

scale  of  solar  system,  262 

transit  of  Venus,  268 
stellar,  averages,  Kapteyn,  345 

Bessel's,  192 

clusters,  352 

defined, 192,  330 

light-year,  333 


measurement  of,  331 
orbits,  331 

photographic  method,  332 
relative,  332 
Parallel  sphere,  41 
Pendulum,  Foucault's  experiment,  89 

Richer's,  at  Cayenne,  98 

shape  of  earth  from,  101 
Penumbra,  eclipse  shadow,  303 
Perigee,  lunar,  169 

tides,  254 
Perihelion,  defined,  120 

earth  in,  123 

passage,  time  of,  201 
Period,  direct  observation  of  planetary, 
192,  215 

Draconitic,  305,  417 

Saros  in  eclipses,  304 

planetary,  in  Kepler's  laws,  188 
orbital  element,  201 

sidereal  and  synodic,  lunar,  161 
planetary,  208,  408 

table  of  approximate  planetary,  211 
Periodic  changes  in  earth's  orbit,  125 
Periodic  comets,  313 
Periodic  perturbations,  206 
Periodic  variable  stars,  328 
Periodicity,  magnetic  storms,  etc.,  290 

meteor  showers,  316 

recurrence  of  eclipses,  304 

sunspots,  290 
Perpetual  calendar,  147 
Perseid  meteors,  316 
Personal  error,  266 
Perturbations,  by  stars,  322 

by  comets,  308 

of  planetary  orbits,  206 

of  planetoid  orbits,  235 

solar  parallax  from,  271 
Peru,  terrestrial  arc  measured  in,  99 
Phase,  11 

lunar,  162 

effect  on  tides,  255 
in  eclipses,  301 

Mars,  222 

Mercury,  218 

Saturn's  ring,  242 

Venus,  218 
Phobos,  inner  satellite  of  Mars,  222 

in  nebular  hypothesis,  357 
Photographic  observations,  Mars,  229 

planetoids,  234 

stellar  magnitudes,  325 

stellar  parallax,  332 

stellar  spectra,  336 
Photographic  telescopes,  281 


430 


INDEX 


Photometer,  stellar,  324,  418 
Photosphere,  solar,  288 
Piazzi,  discovers  Ceres,  232 
Pickering,    spectroscopic    binary    stars, 

349 

Plane,  invariable,  207 
Planetesimal  hypothesis,  358 
Planetoids,  183,  196,  231 

Ceres,  the  first  one,  232 

Eros,  236 

mass  and  size,  235 

Pallas,  Juno,  Vesta,  233 

Wolf,  photographic  discovery,  234 
Planets,  axial  rotation  period,  202 

brilliancy,  46 

curves  in  motion  of,  215 

disks  visible  in  telescope,  13 

elongation  from  sun,  211 

field-glass  view  of,  52 

identification  of,  47 

inferior,  209 

mass  measured,  204,  405 

morning  and  evening  stars,  22 

motion  among  stars,  10 

•names,  10,  183 

near  ecliptic  always,  47 

not  self-luminous,  11 

on  meridian  at  midnight,  52 

opposition  of,  212 

orbital  elements,  200 

orbits  determined,  197 

oscillations  of,  213 

periods  in  Kepler's  laws,  188 

phases,  11 

planetesimal  hypothesis,  359 

proximity  to  us,  10 

revolution  around  sun,  10,  183 

rotation  poles  of,  203 

retrograde  motions,  214 

sidereal  period,  207,  408 

size  measured,  203 

superior,  209 

surface  and  volume  measured,  204 

synodic  period,  207,  408 

twinkling,  50 

ultra-neptunian,  249 

visibility  of,  211 
Planisphere,  63 
Pleiades,  motion  in,  336 
"Pointers,"  constellation,  see  Ursa  Major 
Polar  axis,  in  telescope  mounting,  280 
Poles,  celestial  and  terrestrial,  32 

motion  of,  seen  by  travelers,  39 

rotation  of  celestial,  132 

of  ecliptic,  131 

planetary,  position  of  rotation,  203 


position  of,  above  horizon,  40,  365 

precessional  motion  of  celestial,  131 
Pole  star,  effect  of  precession  on,  132 

how  to  find,  53 

Position  angle,  binary  stars,  348 
Power  of  telescope,  light-gathering,  275 

magnifying,  273 
Precession  of  equinoxes,  126 

cause  of,  129 

changes  right-ascension,  etc.,  334 

determines  date  of  pyramids,  133 

effect  on  pole  star,  132 
Prime  meridian,  Greenwich,  34,  73 
Prism,  in  spectroscope,  282 
Proctor,  motion  of  "Dipper"  stars,  335 
Prominences,  solar,  293 
Proper  motion,  of  stars,  334 

determines  apex,  339 

in  Pleiades,  351 
Ptolemy,  phases  of  Venus,  219 

planetary  theory,  189 
Pyramid,  date  of  construction,  133 
Pythagoras,  earth's  motions,  87 

Radial  velocity,  stellar,  334 

determines  apex,  338 
Radiant,  of  meteor  showers,  315 
Radiation  of  heat  from  earth,  122 
Radius  vector,  119 

law  of  areas,  120,  184 
Rate  of  chronometers,  157 
Recurrence,  of  eclipses,  304 

of  meteor  showers,  316 
Refraction,  atmospheric,  114 

correction  of  sextant  observations,  156 
"Regulator"  clocks,  279 
Relative  stellar  parallax,  332 
Retrograde  motions  of  planets,  214 
Reversing  layer,  in  solar  spectrum,  288 
Richer,  Cayenne  observations,  98 
Right-ascension,  34,  363 

measured,  278 

of  meridian,  67,  366 

sidereal  time  and  hour-angle,  367 
Right  sphere,  40 
Rigidity  of  earth,  112 
Ring  nebulae,  353 
Ring  of  Saturn,  13,  241,  412 

constitution  of,  245 

disappearance  of,  244 

Keeler  and  Maxwell,  245 

phases,  242 
Rising  and  setting,  31 

heliacal,  127 

of  moon,  176 
Roemer,  observes  Jupiter's  satellites,  239 


431 


INDEX 


Rotation,  axial,  celestial  sphere,  30 

earth,  15,  30 

Foucault  experiment,  89 

Jupiter,  236 

Mars,  221 

Mercury,  218 

moon,  169 

Newton's  experiment,  91 

planets,  202 

position  of  poles,  203 

sun,  295 

tidal  effect  on  earth's,  253 

Venus,  221 
Runaway  star,  335 

Sagredus,  character  in  Galileo's  Dialogue, 

88 

Salusbury,  translator  of  Galileo,  88,  237 
Salviati,  character  in  Galileo's  Dialogue, 

88 

Sappho,  planetoid,  267 
Saros,  eclipse  period,  304 
Satellites, 

distance  from  planets,  205 

eclipses,  238 

in  planetesimal  hypothesis,  360 

Jupiter,  237 

longitude  from  observing  them,  239 

Mars,  222 

Saturn,  246 
Saturn,  appearance  in  telescope,  241 

how  to  find,  51 

moons,  246 

ring,  13,  241,  412 
Scale,  of  solar  system,  262,  381 

of  stellar  system,  346 
Schehallien,  mountain  in  Scotland,  104 
Schiaparelli,    meteor   and    comet  orbits, 

319 

Schwabe,  periodicity  of  sunspots,  290 
Scintillation,  see  Twinkling 
Scorpius,  constellation,  diagram,  62 
Seasons,  explanation,  44,  120 

Mars,  222 

Mercury,  217 

Jupiter,  237 

Secchi,  stellar  chemistry,  337 
Secular  perturbations,  206 
Seeing,  process  of,  228 
Semi-diameter,  in  sextant  observing,  156 
Semi-diurnal  tides,  253 
Sextant,  in  navigation,  152 

theory  of,  393 

Shadow,  of  earth  in  eclipses,  303 
Shooting  stars,  see  Meteors 
Showers  of  meteors,  315 


Sidereal, 

day,  65 

period,  moon,  161 
planets,  207,  408 

space,  unit  of,  345 

time,  65,  366 

year,  128 
Sight  line,  of  telescope,  277 

stars'  motion  in,  334 
Sirius,  magnitude,  324 

member  of  Dipper  group,  336 

velocity  observed  by  Huggins,  336 
Sky,  color,  113 

definition,  22 
Slough,  Herschel  at,  247 
Solar  system,  chance  of  reaching  Vega, 
341 

cosmic  motion,  338,  419 

future  of,  361 

in  planetesimal  hypothesis,  360 
Solar  time,  see  Apparent  solar  time 
Solstice,  observed  to  get  length  of  year, 
126 

summer,  93,  121 

winter,  121 

Sosigenes,  arranges  Julian  calendar,  138 
Southern  hemisphere,   conditions  there, 

43,  122 
Space,  2 

unit  of  sidereal,  345 
Specific  gravity  of  earth,  110 
Spectroheliograph,  294 
Spectroscope,  282 

chemistry  of  sun  and  stars,  284 

Doppler  principle,  284 

radial  velocities,  335 

slitless,  285 

solar  prominences,  293 

used  for  binary  stars,  349 
Mars,  224 
Nebulae,  4 
Saturn's  ring,  245 
Spectrum, 

bright-line  and  continuous,  283 

classification  of  stellar,  337 

cometary,  308 

flash,  in  solar  eclipses,  288 

Fraunhofer  lines  in  solar,  287 

reversing  layer  in  solar,  288 

shift  of  lines  in,  284 
Sphere,  celestial,  23 

apparent  rotation,  30 

oblique,  42 

parallel,  41 

right,  40 

crystal,  in  Ptolemaic  theory,  189 


432 


INDEX 


Spiral  nebula?,  5 

Spots  on  the  sun,  17,  289 

Spring  tides,  255 

Stability,  of  planetary  orbits,  206 

Stadium,  Greek  linear  measure,  94 

Standard  clocks,  astronomic,  278 

Standard  magnitudes,  stellar,  324 

Standard  meridians,  74 

Standard  pound,  102 

Standard  time,  65,  74,  83 

Stars,  analogy  to  sun,  6,  322 

artificial,  for  photometry,  325 

average  distance  asunder,  346 

binary,  347 

change  in  distance  of,  spectroscopic,  284 

chemistry  of,  spectroscopic,  284 

clusters,  351 

collision,  347 

community  of  motions,  335 

cooling,  6 

cosmic  velocity,  346 

dates  when  on  meridian  at  9  P.M.,  58 
rising  and  setting,  9  P.M.,  60 

density  of,  in  sidereal  space,  345 

distance,  330 

double,  8 

excessively  remote,  322 

faintest  visible  in  telescope,  418 

fixed,  7,  333 

heat  of,  326 

Herschel's  comets,  355 

Huggins,  their  chemistry,  337 

identifying  them,  52 

Kapteyn's  statistical  researches,  342 

kinetic  theory  of,  347 

light-ratio  of,  324 

line-of-sight  motions,  334 

list  of  brightest,  57 

magnitudes,  6,  323 

maps,  45 

Milky  Way,  354 

morning  and  evening,  22 

novce,  or  new  stars,  8,  327 

number  visible  to  eye,  7 

originate  in  nebulae,  4 

parallax,  330 

periodically  variable,  328 

points  of  light  only,  12 

proper  motions,  334 

radial  velocity,  334 

runaway,  335 

self-luminous,  6 

spectra  photographed,  336 

streams  of,  347 

subject  to  gravitation,  7 

total  light  of,  325 


twinkling,  6 

variation  of  brightness,  8,  326 

in  clusters,  351 
Stationary  points,  in  planetary  motion, 

215 

Statistics  of  stars,  342 
Storms,  magnetic,  periodicity,  290 

solar,  289 

Streams,  stellar,  347 
Summer,  heat  of,  120 

longer  than  winter,  123 
Sun,  absorption  in  outer  layers  of,  286 

analogy  to  stars,  6,  322 

angular  diameter,  118 

annular  eclipses,  304 

apex  of  its  cosmic  motion,  338 

axial  rotation,  17,  295 

central,  355 

chemistry  of,  286 

chromosphere,  293 

corona,  295 

density,  292 

dimensions,  17,  290 

direction  of  rotation  axis,  296 

distance,  from  Jupiter's  satellites,  240 
scale  of  solar  system,  262 
see  Parallax 

eclipses,  297 

eclipse  limits,  300 

effect  on  tides,  254 

faculae,  289 

focus  of  earth's  orbit,  116 

Fraunhofer  lines,  287 

gravity  on,  102,  291 

heats  earth,  121 

mass,  291,  415 

motion,  in  ecliptic,  29 
in  space,  7,  338 

nodules,  289 

photosphere,  288 

planetesimal  hypothesis,  360 

position  on  sky,  25,  27 

prominences,  293 

reversing  layer,  288 

semi-diameter  correction,  156 

source  of  light  and  heat,  292 

spots,  17,  289 

stellar  magnitude,  324,  326 

twin  suns,  347 

volume,  292 
Sundial,  78 

mathematics  of,  368 
Sunspots,  17,  289 

possible  stellar,  328 
Superior,  conjunctions,  210 

planets,  209 


2F 


433 


INDEX 


Surface,  area  of  planets,  204 

of  aerolites,  321 

Syene,  Eratosthenes  observes  at,  94 
Synodic  period,  moon,  161 

planets,  208,  408 

Tail,  comets,  308 

meteors,  315 
Telescope,  272 

cross-threads,  275 

eye-piece,  273 

equatorial,  279 

magnifying  power,  274 

mounting,  276 

object-glass,  272 

photographic,  281 

sweeping  sky  with,  310 
Temperature,  depends  on  day  and  night, 
120 

increase  inside  earth,  111 

Jovian,  237 

Martian,  226,  411 
Temple's  comet,  319 
Temporary  stars,  8,  326 
Terrestrial  telescopes,  273 
Tides,  apogee,  and  perigee,  254 

caused  by  moon,  251 

diurnal  inequality,  253 

effect  of  sun  on,  254 

effect  on  binary  stars,  350 

effect  on  moon,  257 

evidence  of  solidity  of  earth,  111 

inequality  of,  253 

Long  Island  Sound,  256 

oscillations,  253 

planetesimal  hypothesis,  358 

semi-diurnal,  253 

situation  of  crests,  253 

spring  and  neap,  255 

tidal  evolution  and  friction,  256 
Time, 

apparent  solar,  67 

determined  by  observation,  279 

differences,  72 

equation  of,  134 

mean  solar,  71 

sidereal,  65 

standard,  65,  74 

sundial,  82 
Timocharis,  129 
Torsion  balance,  107 

constant  of,  108,  375 
Transit,  of  Mercury,  306 

of  Venus,  221,  268,  306,  412 
Triangulation,  geodetic,  95 
Tropical  year,  128 


length  of,  141 

used  in  calendar,  141 
Tuttle,  comet,  319 
Twilight,  113 
Twinkling,  planets,  50 

stars,  6 

Tycho  Brah6,  Kepler  uses  his  observa- 
tions, 194 

observes  comet  of  1577,  311 

observes  temporary  star,  327 

Ultra-Neptunian  planets,  349 
Umbra,  eclipse  shadow,  303 

sunspots,  290 
Unit,  mass  and  weight,  102 

modern  system,  108 

natural  and  artificial,  140 

sidereal  space,  345 

time,  65 
Universe,  1,  356 
Uranus,  246 

Ursa  Major,  constellation,  community  of 
motion  in,  336 

how  to  find,  52 

Variable  stars,  8,  326       ' 

in  clusters,  351 

periodic,  328 

Variation  of  latitude,  112 
Vega,  distance  of,  420 

light  emitted  by,  418 

near  apex,  338 

possibility  of  solar  system  reaching  it, 

341 
Velocity,  cosmic,  of  solar  system,  338 

of  stars,  346 

of  light,  333 

of  "  runaway  "  star,  347 
Venus,  14,  218 

atmosphere,  220 

attains  greatest  luminosity,  219 

how  to  find,  51 

markings  on,  221 

phases,  218 

transit,  221,  268,  306,  412 
Vernal  equinox,  35,  72 
Vertex  of  angle,  defined,  29 
Vesta,  planetoid,  234 
Victoria,  planetoid,  267 
Visibility,  comets,  311 

of  objects  by  sunlight,  113 

planets,  211 
Vogel,  eclipse  theory  of  Algol,  328 

photographs  stellar  spectra,  336 
Volcanoes,  112 


434 


INDEX 


Volume,  comets,  307 
moon,  173 
planets,  204 
sun,  292 

Wave  motion,  tidal,  255 
Week,  calculation  of  day  of,  143 
Weight,  distinction  from  mass,  102 

of  earth,  103    • 

of  moon,  173 
Winter,  cold  of,  120 

shorter  than  summer,  123 

southern  hemisphere,  124 


Witt,  discovers  Eros,  236 

Wolf,  photographs  planetoids,  234 

Year,  ancient  methods  of  determining  its 

length,  126 
in  chronology,  140 
number  of  days  in  it,  70 
sidereal  and  tropical,  128 
synodic,  208 

Zenith,  36 

stars  brightest  near,  325 
Zero,  a  stellar  magnitude,  324 
Zodiacal  light,  249 


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